There are various things in place to ensure games move quicker than snail pace.
As for ending turns and phases ending, it all depends on the game. Each round has at most 5 phases, regardless of the number of players in the game. Each player is allocated to a phase, and once all the players in a phase have finished their turn, it ticks over to the next phase. Once all phases in a round finish, it goes to the next round, and it all starts again.
Also, if a player starts their turn, they have at most two hours to finish it, otherwise it finishes automatically. This stop players from starting their turning then walking away and causing a 24 hour delay.
The biggest problem on the site is players who join games then don't show up. If a player misses two turns in a row though, they are kicked out of the game. So quite a few games are slow for rounds 1 and 2, then speed up considerably.
There's an FAQ post here
that talks about rounds and phases, that should answer most of your questions. Note that games with 5 teams (1 player per team) or any team game with 5 players per team is the slowest, with 5 phases per round.
What game are you referring to, if I may ask? I'll give you a quick breakdown of the rounds and phases to demonstrate what is happening.