Just played multiple games simultaneously without logging out of the game, and continuously got emails telling me a new turn was ready, which I already saw in-game. It would be nice if the email notification for new turns is turned off while you are in the game itself, as I now have a ton of emails telling me about turns I already took.
Second comment is that I just lost a game, and it immediately took me back to the menu. However, I could see there was a message from the other player, but was unable to read it due to automatically leaving the game and after that I could not return to it as it was finished.