Learn Turkish: Banyoda çabuk yıkandılar ve tarandılar. - They washed and combed their hair quickly in the bathroom.

It makes the verbs reflexive, meaning the subject does the action to themselves. Yıkamak means to wash something else, but yıkanmak means to wash oneself (take a shower or bath). Likewise, taramak is to comb something, but taranmak is to comb one's own hair.

The English translation says they combed their hair, but I don't see the word for hair (saç). Why is it missing?

Because the reflexive verb taranmak literally means to comb oneself. Since hair is the only thing you would normally comb on yourself, the word saç is naturally implied in Turkish and doesn't need to be spoken.

Why do both verbs have the plural ending -lar (yıkandılar, tarandılar)?

They both have -lar because they both share the hidden subject they (onlar). However, in Turkish, you can actually drop the person ending on the first verb when using ve (and). Saying yıkandı ve tarandılar would also be completely correct and very common!

Does the word çabuk (quickly) apply to both washing and combing?

Yes! Because çabuk is placed right before the pair of verbs connected by ve, it acts as an adverb for both actions. They did both the washing and the combing quickly.