Tu me fais me sentir is the right translation of “you make me feel“.

Tu me fais sentir is quite odd

Tu me fais sentir comme un roi

would mean

You make me smell like a king

See La différence entre « sentir » et « se sentir »

You have to make the difference between “se sentir” and “sentir”.

Se sentir : Avoir conscience d’être dans tel ou tel état (to feel)

Sentir :

1. Percevoir quelque chose par l’odorat (to smell)
2. Éprouver dans son corps les manifestations, les effets de quelque chose (to feel)

In most cases, the common usage is to say: “Tu me fais me sentir” for “you make me feel“. Saying “tu me fais sentir” could be misinterpreted as “you make me smell (the flower for example)”.

Orally, it happens that the “me” is very little pronounced : “Tu m’fais m’sentir” which can be confused with “Tu m’fais sentir”.

However, in some contexts, it is okay to say “tu me fais sentir roi” if feeling king is not really a state but a deduced sensation.

For example : “Tu me fais sentir femme” is common expression. In the case where the sentence is pronounced by a woman, there is no doubt about her being a woman, but she expresses the sensation, the effects of being a woman. That is why she can use sentir as to feel.

There is no grammatical rule involved and this is plainly a matter of meaning. You are dealing with two verbs, each with its meaning, one of the type called “pronominal” (se sentir) and the other “a plain transitive verb” (sentir quelque chose ou quelqu’un).

I

“se sentir” means “to feel oneself”

example: I felt myself being pulled down towards the bottom of the river.

Cela se traduit par “Je me sentais tiré vers le fond de la rivière.”

dictionary: Avoir conscience de l’état physique ou moral dans lequel on se trouve

II

“sentir quelque chose ou quelqu’un” has two main meanings, one is literal, the other figurative.

example:

-literal- I could smell flowers and the scent of freshly cut grass.

This is translated as “Je sentais des fleurs et et l’odeur de l’herbe qui venait d’être coupée.”.

meaning (a): to smell

-figurative- The rider felt the horse was getting tired and slowed down its pace to a trot.

A translation of this is “Le cavalier sentit que le cheval était fatigué et le mis au trot.”.

meaning (b): to feel

III

In the first case (tu me fais me sentir) the verb is the pronominal verb; the translation is “you make me feel” and in English you do not use a pronominal form (you make me feel myself).

For instance “He made me feel forlorn.” will be in French “Il me faisait me sentir délaissé(e).”.

Another example: They make her feel as if she were the worse possible candidate for the post.

A translation of that could be “Ils la font se sentir comme si elle était la pire des candidates pour le poste.

It is very interesting here to remark the different pronoun used if the other verb is found; there is next an example of that.

The new clothes she bought make her feel the curves of her body, or so says she.

The translation is now “D’après elle, les nouveaux vêtements qu’elle a achetés lui font sentir les courbes de son corps.”

“se sentir les courbes” would not be correct in this sentence.

dictionary: Percevoir, éprouver une sensation physique qui renseigne sur l’état de l’organisme ou sur le milieu extérieur

example: They make you feel much less the burning of the sun’s rays.

A translation of that could be “Ils vous font sentir beaucoup moins la brulure des rayons du soleil.

In the second case you are dealing with the figurative verb, in “II“.

example: The festivities made her feel a certain emptiness in her soul, gave her a sense of her being useless.

In French we have “Les réjouissances lui faisait ressentir une certaine absence d’âme, lui donnait une impression d’être inutile.

For this latter translation “ressentir” is idiomatic whereas “sentir” is not quite correct but the idea is close and can be understood.

One must say that there is not too great a difference between “Avoir conscience de l’état physique ou moral dans lequel on se trouve” and “Percevoir, éprouver une sensation physique qui renseigne sur l’état de l’organisme ou sur le milieu extérieur” but it is important enough to compel one to discriminate between the two.