Where for most situations there is no use, we intentionally added the turn marker. Players could ignore this as they wish, just like any group would use their own rules for any game. Feel free to leave the turn marker in the box as you like.
However, there are some uses for it. First players may want to know what's the 'perfect progression' on the AST. The leader may remember he just froze on the AST once, but after a few hours, some players that have stayed on the AST for a few turns may want to see how many turns they have fallen back, or how many turns the leader is away from perfect progression. Especially in games with a large number of players where some players might not keep track of all players at the other end of the board.
Secondly, players may want to set a fixed number of turns to player instead of playing a scenario (short game or first game) or instead of setting a time limit. It it also great for explaining the rules, like using the first game scenario.
In the third place, we've seen a lot of games with experienced players, especially those with a low player count, where a lot of additional turns are played, just because players keep attacking the leader. (i love those games). This might turn out once you're more experienced in MegaCiv. There you want to know the number of turns actually played. I've seen a game of 21 turns, and that wasn't because the players were inexperienced. They were all playing really tough. Using the Expert AST you fall back one space when having 0 cities. I've seen players holding on to Gold cards so that regression falls more often. ouch.