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If some people play a strategic board game for example, and I wonder what a certain player is going to do and what his strategy is, may I say:
What is he going to do on his turn / on the next turn?
Another example:
Somebody is playing a game solo with many turns (?) on the computer and the programme is always showing him what turn he is on.
If you had to use a preposition, 'on' would be the most common.
On my next turn, I'm going to....
Equally, or perhaps more commonly said would be:
When it's my turn, I'm going to...
If the turn can involve multiple steps or activities, you can also say:
During my next turn, I'm going to...
If you're playing a game where opponents take turns, the preposition "on" is appropriate—on my turn, on his next turn, etc. This is true even if playing against a computer, e.g.Chess or Scrabble.
By your question mark after "turns" in reference to solo computer games, I infer that you might be wondering whether "turn" is the appropriate term for games in which one has multiple distinct opportunities to play, e.g, to accumulate points and/or advance to another level. In many such games, particularly shoot-em-up (combat) games, it's a do-or-die situation in which your "turn" ends when you "die" and you get another "turn" only if, like the proverbial cat, you have more lives to use up.
In these cases, I would say that "turn" is NOT appropriate. The computerized antagonists are not "taking "turns", so the solo player is not "taking turns". The separate periods of play could be called rounds, or lives.
On is the right preposition for turn when it means time for one to make a move in a game.
Regarding traffic/driving, sometimes an intersection where a road turns left or right is called a turn. You use at here as is typical for a place.
Wait for the light at the turn, then once you're on 2nd Avenue look for the red building.
In turn is a well-known phrase that means one after another.
The pack of ill-behaved boys all threw rocks at the windows in turn.
