Do you have a password set through the website?
Sounds like the player before you might be taking the beginning of your turn.

@Xaviien I do not think that is possible, as the GMR will not let you upload a half played turn. At least that is what I experienced when I mistakenly tried to upload a save file, in which my turn was half played (had units that need to move and / or city production to choose etc.), GMR gave me an error message telling me that I attempted to upload an unfinished turn.

OK, I do vaguely remember that may have happened once before, now that you mention it.
I'm just throwing shit at the wall, to see what sticks, like a doctor.

@glacierslayer In one of my games, at the start of my turn, I had some units that were out of moves. Another player told me that reloading the save file would fix the issue. It did not work for me, but maybe it will work for you.

There's a bug in Civilization V where keystrokes can still effect the game when typing your password. I'd recommend removing your in-game password on GMR – we make your turn unplayable to other players whether or not your password is set anyway.