The driver's side and passenger side front floor of my 2003 Corolla started getting very wet a few weeks ago. When I make sharp turns I can sometimes hear a sloshing sound from the right front. I can't find any obvious leaks. Can anyone suggest places to look and ways to locate the leak? Many thanks

You have a plugged air conditioner evaporator coil drain. Take a wire or very long pipe cleaner and push in up into the drain tube underneath the car. It's likely located along the fire wall, on the passenger side. Water should gush out. It's a 2 minute job.

Txgreasemonkey is the expert here, and doesn't need my backup... just wanted to mention that what you experienced and what he mentioned happened to me with my nissan.

Has this just started since you've been running the air conditioner? That's what happened in mine. Then id turn a corner and water would drain out into the passenger side floor.

Just to add, agree with all the above. Be prepared for a gallon of water to pour out once that drain obstruction is removed.
May want to have a catch pan ready if you're in the garage on car ramps or something like that. Good luck!