Learn Hardcore Turkish: Sıcak suyu kapatmak için vanayı yavaşça çevir. - Turn the valve slowly to shut off the hot water.

The -yu part is actually the accusative (definite object) suffix -u, which marks suyu as a specific “hot water” you want to shut off. Because su ends in a vowel, Turkish inserts a buffer consonant -y- before the suffix to avoid two vowels running together. So su + y + u → suyu.

Why does vanayı also have an -ı suffix?

Similar to suyu, vana (the valve) is the definite direct object of the main verb çevirmek. To show it’s a specific valve you’re turning, you add the accusative suffix -ı. Because vana ends in a, vowel harmony dictates -ı; and since a is a vowel, you again need the buffer -y-: vana + y + ı → vanayı.

What does -mak için do in this sentence?

-mak için attaches to a verb infinitive to form a purpose clause meaning “in order to ….” Here kapatmak için means “in order to shut off.” So Sıcak suyu kapatmak için = “in order to shut off the hot water.”

Why is the verb command given as çevır and not çevirmek or çevırın?

Çevirmek is the infinitive (“to turn”).
For a singular, informal command you drop -mek/-mak entirely: stem çevir.
If you wanted a polite or plural command, you’d use the 2nd-person plural imperative çevirin.

How is the adverb yavaşça formed, and how does it differ from just yavaş?

Yavaşça is formed by adding the adverbial suffix -ça to the adjective yavaş (“slow”). Both yavaş and yavaşça can mean “slowly,” but yavaşça is more explicitly an adverb. You’ll often hear both in spoken Turkish, though yavaşça is very common when giving instructions.

Why is the subject not mentioned in this sentence?

In Turkish imperatives, the subject “you” is always implied and dropped. Verb forms themselves show you’re addressing “(you) turn …” so there’s no need for a pronoun.

How can the sentence have two objects—sıcak suyu and vanayı? Which object goes with which verb?

This is a two-clause structure:
1) Purpose clause: “sıcak suyu kapatmak” – here sıcak suyu is the object of kapatmak.
2) Main clause: “vanayı yavaşça çevir” – here vanayı is the object of çevirmek (in the imperative form).
Each clause takes its own accusative suffix.

Why is the infinitive kapatmak and not kapatmek?

Turkish infinitives follow vowel harmony: if the last vowel of the root is a, ı, o, u, the infinitive suffix is -mak; if it’s e, i, ö, ü, the suffix is -mek. Since kapat- has an a, you use -mak.

Can I move yavaşça to another position in the sentence?

Yes. Turkish word order is relatively flexible. The most natural is vanayı yavaşça çevir (object–adverb–verb), but you could say yavaşça vanayı çevir for emphasis on slowness or even vanayı çevir, yavaşça in spoken style. However, placing the adverb immediately before the verb is most common.

Is this command polite, neutral, or informal? How can I make it more polite?

As written, the command “vanayı yavaşça çevir” is neutral–informal (talking to one person you know well). To make it polite or address multiple people, use the 2nd-person plural imperative:
Sıcak suyu kapatmak için vanayı yavaşça çevirin.
Adding lütfen (“please”) at the start or end also increases politeness.