Although we can sometimes use the noun il turno to mean "the turn," as in, "Wait your turn" (aspetta il tuo turno), there's another (colloquial) expression we use in Italian, more often than not. We use the verb toccare (to touch). In the following clip, Dino and Melody are making wishes with blueberries:
Adesso tocca a te.
Now it's your turn.
Caption 9, Sposami - EP 2 - Part 20Play Caption
Tocca a te (it's your turn).
Tocca a me (it's my turn).
The question you might get in a shop where various people are waiting their turns:
A chi tocca (whose turn is it)?
The answer can be tocca a me, tocca alla signora, tocca a lei, tocca a loro...
Twisting this expression a bit turns it into something you have to do.
Mi tocca (I have to do it).
Ti tocca (you have to do it).
Ho faticato tanto per averla,
I worked so hard to get it,
e adesso mi tocca venderla.
and now I have to sell it.
Captions 6-7, Il Commissario Manara - S2EP10 -La verità nascostaPlay Caption
The important thing to remember in using this expression is that the person is the indirect object. The preposition of choice is a (to, at). The subject is a general "it," implied, or absent, actually.
In some places, you take a number and then wait your turn, at the supermarket, for example, at the bread counter, or the counter where you get prosciutto. Otherwise, you can ask, Chi è l'ultimo (who's the last [in line])?