That sounds very weird.
Several things to check, in some random order.
1) Click on "your settings" (above the thick red line near the top of
the screen) and select the "Go to next waiting game after move"
checkbox - make sure it has a check mark in it.
2) Alternatively, make your move in the current game, then click "your
games" which should give you a list of all your games. Select the one
If those aren't working, something is REALLY weird. System seems to
be working fine for me, and I presume others too.