A short delay is possible and expected when playing Panda Poet. This is because the game only checks intermittently to see whether your opponent has played a move. If your opponent happened to play a move just before the check happens, you'll see the move almost immediately. If your opponent happened to play a move just after the check, you'll have to wait until the next time the game does its check. This wait shouldn't be more than a minute or so if you've been actively playing back and forth; however, the checks do slow down if the game has been left open in the browser window a long time without any activity, so if you've left the game up for a long time then it might be several minutes between when a move is played and when it shows up on the other side.
If you know that your opponent has taken his/her turn (e.g., perhaps you were chatting on the phone and he/she told you so), you can force the check to happen by exiting to the main menu and then re-entering the game you were playing. This can move a game along slightly faster. Note that forcing the check to happen won't make the game go faster if your opponent hasn't actually played his/her move!