How does Forgotten Monument work? Is it the 3rd total turn or the third turn for each player? Aka turn 6 total?

According to the card text of Forgotten Monument and the associated RiftJudge FAQs, the restriction "until their third turn" refers to the third turn taken by that specific player, not the total number of turns taken in the game.

Here is how it works:

• Individual Turn Count: You can score the battlefield as soon as it is your own third turn. For example, if you are the first player, your third turn is the 5th turn of the game overall (Player A, Player B, Player A, Player B, Player A).
• Scoring Timing: You do not have to wait until the end of your third turn. As soon as you reach your third turn, the condition is met, and you may score the battlefield immediately if you satisfy the other requirements for scoring.
• Conquering Restriction: Per RiftJudge FAQ #9332, you cannot "Conquer" the Forgotten Monument before your third turn. Because conquering is a method of scoring, you are prohibited from performing that action there until the condition is met.

In summary: It is your third turn, not the sixth turn of the game.