Critics say the Manusmriti is pro-Brahmin, and written by casteist Brahmins by favoring particular section of society and thereby oppressing Shudras, women, and other lower castes.

Is this argument valid?

Yes, it appears so. This is what Patrick Olivelle says in the Introduction to his translation of Manusmṛti (a.k.a Mānava-Dharmaśāstra):

Reading the MDh one cannot fail to see and to feel the intensity
and urgency with which the author defends Brahmanical privilege.
A major aim of Manu was to re-establish the old alliance between
priesthood and royalty, an alliance that in his view would benefit
both the Brahmin and the king, thereby re-establishing the Brahmin
in his unique and privileged position within society. We hear the
repeated emphasis on the inviolability of the Brahmin in his person
and in his property. He has immunity from the death penalty, from
taxes, and from the confiscation of his property.

The king is advised
repeatedly that a Brahmin's property is poison. Stealing a Brahmin's
gold is one of the five grievous sins, and the death penalty is imposed
on the perpetrator. Devotion to Brahmins is a cardinal virtue of
kings: 'Refusal to turn back in battle, protecting the subjects, and
obedient service to Brahmins—for kings, these are the best means of
securing happiness' (7.88). The reason why foreign ruling classes,
such as the Greeks, Śākas, Persians, and Chinese, have fallen to the
level of Śūdras, once again, is their lack of devotion to Brahmins: 'By neglecting rites and by failing to visit Brahmins, however, these men of Kṣatriya birth have gradually reached in the world the level of Śūdras' (10.43).

The Brahmanical privilege is threatened from two quarters: the
Śūdra, within which class Manu often lumps all the lower classes of
society, and the Mleccha (foreigner, barbarian). Now, it is true that
even the Dharmasūtras contain passages that are anti-Śūdra. It is
taken for granted that the sole duty of Śūdras is to serve the upper
classes; penalties for killing a Śūdra are much less than for killing
people of the upper classes; likewise, penalties are increased for
guilty Śūdras; the list could go on. Yet we also see that Śūdras acted
as cooks in Brahmin households Āpastamba (2.29.11-15) even says
that one may learn aspects of the Law (dharma) from Śūdras. There
is a virulence in Manu's rhetoric vis-a-vis Śūdras lacking in the
Dharmasūtras that appears to indicate that there must be a subtext to
it. How could the lowest class of society with little access to material
resources pose such a threat to social order and to Brahmanical
hegemony? The fear of the Śūdra contrasts sharply with Manu's
view of Vaiśyas. These are dealt with in a dispassionate and straightforward way. Why were Vaiśyas, who are depicted as agriculturalists
and traders, that is, people with resources, not a threat to the Brahmin-Kṣatriya alliance that Manu was attempting to forge and strengthen? At one level, I think, historical memory is at work here;
Śūdras were once in power and posed a real threat to Brahmanical
hegemony, and history can always repeat itself.

Beyond that, however, 'Śūdra' for Manu is often a code word; it
identifies the enemy and it encompasses a wide cross-section of society, both past and present. It evoked the memories of bad old days; it
heightened the anxiety that what happened under the Mauryas
could be repeated. I also think that there was a contemporary threat
to Brahmanical supremacy not so much from political power but
from rival religious establishments, especially the Buddhist and the
Jain monastic orders. I think Manu includes these within his code
'Śūdra'. The connection between Śūdra and the non-Brahmanical
ascetic sects is drawn by Manu himself. In his advice regarding a
Brahmin's residence, Manu (4.61) says: 'He should not live in a
kingdom ruled by a Śūdra, teeming with unrighteous people, overrun by heretical ascetics, or swamped by lowest-born people.' Here
we have a clear juxtaposition between a kingdom ruled by a Śūdra king and a region populated by heretical ascetics (principally, Buddhists and Jains), by lowest-born people, and by unrighteous men.

Indeed, Manu's instruction (9.225) to the king about cleansing his
kingdom of dangerous people includes men who belong to heretical
sects. The strength of Buddhism in the north-western regions during this period and the patronage offered to them by what for Manu were Mleccha (foreign, barbarian) kings may also have influenced the
connection between heretic and Śūdra/Mleccha.

Alongside Śūdras, we have the Mlecchas. Manu is cognizant of
the regions occupied by the foreign barbarians, for at 2.23 he defines
the areas outside the central Aryavarta as the region of Mlecchas.
Manu, however, does not have much to say about the Mlecchas in the
rest of the book; his focus is on the Śūdras. Or, is the code 'Śūdra'
meant also to encompass these other outsiders as well?

Note that at 10.44 Manu presents the Mleccha groups such as Greeks, Śākas, and
Chinese as sunk to the level of Śūdras, although they were Kṣatriyas by birth.

The ideology that drives Manu explains the plan of his book. He
devotes 1,034 verses (38.6 per cent) to the discussion of the Brahmin
and 971 verses (36 per cent) to matters relating to the king; these two
take up three-quarters of the entire text.

Manu's agenda is twofold:

• he wants to tell Brahmins how to behave as true Brahmins devoted to Vedic learning and virtue, and


• he wants to tell kings how to behave as true kings, devoted to Brahmins and ruling the people justly.

For this agenda he brings the authority of no less a person than the Creator
himself, who is presented as the absent author of the text.

I think the following prescriptions of Manu taken from https://vedkabhed.wordpress.com/2015/05/01/manu-smriti-and-untouchables/, IF QUOTED AND TRANSLATED CORRECTLY, Do show that these were in fact written by the castiest brahmins to oppress sudras. Its beyond my understanding how an ordained and illumined soul can prescribe punishments with so much of hatred and cruelty. Otherwise we must treat these verses as interpolations. (But who would decide how much is interpolation and how much is original?)

Manu Smriti 5.140 Following the path of equity, Sudras must shave their heads once, each month, follow the rules of purification laid down in respect of the Vaishyas, and eat the leavings of Brahmana’s food.

Manu Smriti 10.129 No collection of wealth must be made by a Sudra, even though he be able (to do it); for a Sudra who has acquired wealth, gives pain to Brahmanas food.

Manu Smriti 10.96 A man of low caste who through covetousness lives by the occupations of a higher one, the king shall deprive of his property and banish.

Manu Smriti 8.270-1. A once-born man (a Sudra), who insults a twice-born man with gross invective, shall have his tongue cut out; for he is of low origin. If he mentions the names and castes (gati) of the (twice-born) with contumely, an iron nail, ten fingers long, shall be thrust red-hot into his mouth.

Manu Smriti 8.272 If a Sudra insolently gives any religious or moral advice to a Brahmana, the king, shall cause hot oil to be poured into his mouth and ears.

Manu Smriti 8.281-2 A low-caste man who tries to place himself on the same seat with a man of a high caste, shall be branded on his hip and be banished, or (the king) shall cause his buttock to be gashed.

Manu Smriti 3.112 Even a Vaisya and a Sudra who have approached his [Brahmin] house in the manner of guests, he [Brahmin] may allow to eat with his servants, showing (thereby) his compassionate disposition.

Manu Smriti 3.92 Let him gently place on the ground (some food) for dogs, outcasts, Kandalas [Chandal] (Svapak), those afflicted with diseases that are punishments of former sins, crows, and insects.

Manu Smriti 5.131 Manu has declared that the flesh (of an animal) killed by dogs is pure, likewise (that) of a (beast) slain by carnivorous (animals) or by men of low caste (Dasyu), such as Kandalas.

Manu Smriti 8.279-80 With whatever limb a man of a low caste does hurt to (a man of the three) highest (castes), even that limb shall be cut off; that is the teaching of Manu. He who raises his hand or a stick, shall have his hand cut off; he who in anger kicks with his foot, shall have his foot cut off.

Manu Smriti 4.80 Let him [i.e., Brahmin] not give to a Sudra advice, nor the remnants (of his meal), nor food offered to the gods; nor let him explain the sacred law (to such a man), nor impose (upon him) a penance.

Manu Smriti 4.140 Let him [i.e., Dvija] not journey too early in the morning, nor too late in the evening, nor just during the midday (heat), nor with an unknown (companion), nor alone, nor with Sudra.

Manu Smriti 5.104 Let him not allow a dead Brahmana to be carried out by a Sudra, while men of the same caste are at hand; for that burnt-offering which is defiled by a Sudra’s touch is detrimental to (the deceased’s passage to) heaven.

NOTE: Manusmriti critics have cherry-picked certain verses from the Manusmriti to show that Manu was unfair towards Shudras and partial towards Brahmins. The purpose of this answer is to provide the rest of the other verses that censure bad Brahmins and praise good Shudras, to show that the critics cherry-picked those verses unfairly, and that Manu is actually fair towards all castes.

In reality, the Manusmriti censures bad Brahmins and bad Shudras, and praises good Brahmins and good Shudras.

Is the Manusmriti pro-Brahmin and written by casteist Brahmins to oppress Shudras?

No, because upon a closer look at the Manusmriti, its intention is to secure the welfare of all living beings.

If the Manusmriti is pro-Brahmin, then how could it have verses like this?

8.102 - ‘He shall treat like Śūdras the Brāhmaṇas who tend cattle, who engage in trade, and who are craftsmen, actors, menial servants or money-lenders.’

(Brâhmanas) who neither study nor teach the Veda nor keep sacred fires become equal to Sûdras;

And they quote a verse of Manu on this (subject), 'A twice-born man, who not having studied the Veda applies himself to other (worldly studies), soon falls, even while living, to the condition of a Sûdra, and his descendants after him.'

'(A twice-born man) who does not know the Veda (can)not be (called) a Brâhmana, nor he who lives by trade, nor he who (lives as) an actor, nor he who obeys a Sûdra's commands, nor (he who like) a thief (takes the property of others), nor he who makes his living by the practice of medicine.'

2.103 - But he [the Dvija] who does not stand during the morning-twilight, and who does not sit through the evening-twilight, should be excluded, like the Sūdra, from all that is due to twice-born persons.

11.90 - A twice-born person, having, through folly, drunk wine, shall drink wine red-hot; he becomes freed from his guilt, when his body has been completely burnt by it.

Gautama (23.1).—‘They shall pour hot wine into the mouth of a Brāhmaṇa who has drunk wine; he will be purified by death.’

Baudhāyana (2.1.18, 19, 21).—‘If he [a Brahmana] has drunk Surā he shall scald himself to death with hot wine.

3.133 - As many mouthfuls as the person [Brahmana] ignorant of the Veda swallows out of the offerings to gods and Pitṛs [at a Sraddha], so many flaming spikes, spears and iron-balls does the man [Brahmana] swallow after death.

Hārīta (Do.).—‘Even those born of noble families and endowed with learning,—if they be of base conduct and addicted to wicked deeds,—they are even regarded as demons. Those addicted to the killing of birds, fish and deer, serpents and tortoise and other animals are all Bad Brāhmaṇas. Who serves a Śūdra, who is supported by the King, the village-sacrificer, those living by killing and capturing—these six are Low Brāhmaṇas.’

5.19 - The mushroom, the village-pig, garlic, the village-cock, onions and leeks,—the twice-born man eating these intentionally would become an outcast.

5.53 - In normal times the twice-born man conversant with the law shall not eat meat unlawfully; having eaten it unlawfully, he shall, after death, be devoured by them helplessly.

5.35 - But when invited according to law, if a man [Brahmana] does not eat meat, he becomes, after death, a beast, during twenty-one births.

And many more verses. Of course, there are many verses praising Brahmanas, but as shown above, there are many verses deprecating bad Brahmanas. So, how can anti-Hindus cherry pick certain verses and portray the Manusmriti as pro-Brahmin? That is unfair, biased, and illogical.

Now let's address another related criticism.

Is the Manusmriti anti-low caste?

No it is not. Dharma is conducive to one's welfare. According to Jaimini's Purva Mimamsa Sutra 1.1.2:

Dharma is that which is indicated by the Veda as conducive to the highest good.

Therefore, how can anyone say that Dharma is wrong or evil?

Here is the Dharma of Shudras:

9.334 - For the Śūdra the highest duty conducive to his best welfare is to attend upon such Brāhmaṇa house-holders as are learned
in the Vedas and famous.

9.335 - If he is pure, attendant upon his superiors, of gentle speech, free from pride, and always dependent upon the
Brāhmaṇa,—he attains a higher caste.

Viṣṇupurāṇa (Parāśaramādhava-Ācāra, p. 419).—‘It is only through attending upon the twice-born that the Śūdra becomes entitled to perform the Pākayajñas; and thereby becoming blessed, he wins the worlds [earth and heaven].—The Śūdra also shall make gifts, and perform the Pākayajña-sacrifices, as also the rites in honour of Pitṛs.’

Why should Shudras serve Brahmanas aside from it being their primary duty? Because according to the Mahabharata:

Mahābhārata—Anuśāsana (Do.).—‘Finding the Śūdra oppressed with bad
traits due to the quality of Tamas, Pitāmaha ordained attendance upon
the twice-born as his duty. Through his devotion to the twice-born,
the Śūdra drops off all those traits due to the quality of Tamas; and
by attending upon the twice-born, the Śūdra attains the highest
good.—Harmless, devoted to good deeds, worshipful towards gods and the
twice-born, the Śūdra becomes endowed with all the rewards of Dharma.’

Shudras are in fact, oppressed by the quality of Tamas, and not by serving Brahmanas! It is by serving Brahmanas that Shudras become Sattvic, and then are no longer oppressed!

Also, some rights given to Shudras that higher castes don't have:

2.23 - But the region where the spotted deer roams by nature is to be known as the ‘land fit for sacrificial acts’; beyond that is the ‘land of the Mlecchas.

2.24 - The twice-born people should seek to resort to these countries [where the spotted deer roams by nature]; the Śūdra may however, when distressed for a living, reside in any land.

10.122 - He shall serve Brāhmaṇas either for the sake of heaven, or for the sake of both; when he has attained the title of the ‘Brāhmaṇa,’ this implies the accomplishment of all his purposes.

10.124 - They [the Dvijas who are served by a Shudra] should provide out of their family, a suitable maintenance for him [the Shudra servant], after considering his own capacity, and the man’s skill and the burden of persons to be supported by him.

10.125 - Remnants of food and worn-out clothes shall be given to him; as also the grain-refuse and old furniture.

10.126 - For the Śūdra there is no sin; nor is he worthy of any sacraments; he is not entitled to any sacred rites; but there is no prohibition against sacred rites.

10.127 - If those [Shudras] who, knowing their duty, and wishing to acquire merit, imitate the practices of righteous men, with the exception of reciting the sacred texts, they incur no guilt; they obtain praise.

11.93 - Wine [Sura] indeed is the dirty refuse of grains, and sin also is called ‘dirt’; for this reason the Brāhmaṇa, the Kṣatriya and the Vaiśya shall not drink wine [but the Shudra can].

Gautama 10.67 - If Âryans and non-Âryans interchange their occupations and conduct (the one taking that of the other, there is) equality (between them).

'There is equality between them, i.e. the one need not serve the other. A Sûdra need not serve even a Brâhmana, (much less) any other (twice-born man) who lives the life of a non-Âryan (Sûdra). A Sûdra, even, who conducts himself like an 'Âryan must not be despised by men of other castes, who follow the occupations of non-Âryans, on account of his inferior birth.'--Haradatta.

On the other hand, there are verses that seem to be anti-caste, such as these:

8.270 - If a once-born person insults a twice-born one with gross abuse, he should suffer the cutting off of his tongue; as he is of low origin.

8.271 - If he mentions the name and caste of these men with scorn, a burning iron nail ten inches long shall be thrust into his mouth.

8.272 - If through arrogance, he teaches brāhmaṇas their duty, the king shall pour heated oil into his mouth and ears.

However, these verses stand on the same footing as verses like these:

Gautama (23.1).—‘They shall pour hot wine into the mouth of a Brāhmaṇa who has drunk wine; he will be purified by death.’

11.103 - He who has violated his Preceptor’s bed shall confess his crime and lie down upon a heated iron-bed; or embrace a blazing image. By death he becomes purified.

11.104 - Or, having cut off his penis and testicles, he shall take them in his joined hands and walk straight on towards the ‘region of evil spirits,’ until he falls down [dead].

Therefore, it is not right to select some verses and say "the Manusmriti is anti-Shudra."

Therefore, the Manusmriti was not written by casteist Brahmins who wanted to oppress Shudras.