You can cast Serra Avenger during another player's first, second, or third turns of the game if some other effect (such as Vedalken Orrery) enables that.

If the game is restarted (by Karn Liberated), you can't cast Serra Avenger in your first, second, or third turn in the new game.

Serra Avenger cares about how many turns you have taken, not necessarily how many turns the game has had, in case you take an extra turn.