El chico quiere bañarse.

Because they are two different words:

• el = the (article)
• él = he (subject pronoun)

In El chico quiere bañarse, El is the article belonging to chico, so it means the boy.

A useful detail: even at the beginning of a sentence, él keeps its accent if it is the pronoun. So El chico... is correct, but Él quiere... would mean He wants....

Querer is the infinitive, the dictionary form, meaning to want.
Here the verb has to match the subject el chico, so it must be conjugated:

• yo quiero
• tú quieres
• él/ella quiere
• nosotros queremos
• vosotros queréis
• ellos quieren

Since el chico = he, the correct form is quiere.

Also, querer is a stem-changing verb in the present tense: e → ie in most forms, which is why it becomes quiere, not quere.

Because Spanish works like English here:

• He wants to bathe
• El chico quiere bañarse

After a conjugated verb like quiere, the next verb normally stays in the infinitive:

• quiere comer = wants to eat
• quiere dormir = wants to sleep
• quiere bañarse = wants to bathe

So quiere is the conjugated verb, and bañarse is the infinitive that tells you what he wants to do.

The -se marks the verb as reflexive.

• bañar = to bathe someone/something
• bañarse = to bathe oneself / to take a bath

So bañarse shows that the boy is doing the action to himself.

This is very common in Spanish with actions involving personal care:

• lavarse = to wash oneself
• vestirse = to get dressed
• peinarse = to comb one’s hair
• bañarse = to bathe oneself

Yes. That is also correct.

With a conjugated verb + infinitive, a reflexive pronoun can usually go in either place:

• El chico quiere bañarse
• El chico se quiere bañar

Both mean the same thing.

What you cannot do is use the pronoun twice:

• El chico se quiere bañarse ❌

So you choose one position or the other, not both.

Because in Spanish, a singular countable noun like chico usually needs some kind of determiner:

• el chico = the boy
• un chico = a boy
• mi chico = my boy / my guy

A bare noun like chico quiere bañarse is not normal standard Spanish.

So el chico is the natural way to say the boy.

In Spain, chico is a very common word. Depending on context, it can mean:

• boy
• kid
• young guy
• sometimes simply guy

The feminine form is chica.

Also, chico/chica can be used as adjectives meaning small, but in this sentence chico is clearly a noun: the boy.

A simple English-style guide is:

• quiere ≈ KYEH-reh
• bañarse ≈ bah-NYAR-seh

A few key pronunciation points:

• qu before e or i sounds like k, so quiere starts with a ky- sound.
• The u in qu is silent here.
• ñ sounds like ny in canyon.
• The r in quiere is a light Spanish r, not a strong English r.

So the whole sentence sounds roughly like:

el CHEE-koh KYEH-reh bah-NYAR-seh