I used to use a website where you could chat with people; you entered
the site and chose a cartoon character, then entered an area, eg a
park or a room or a city, and when you typed something it appeared as
a speech bubble. Any ideas what it was?

Many chat sites have been based upon "The Palace," a graphical chat
architecture that uses cartoon avatars and "speech bubbles":

http://www.thepalace.com/

Do you remember what kind of setting your site featured (was it set in
a disco? a space station? a castle?)

Any additional details could make it easier for us to track down the
site you remember.

Was it Habbo Hotel?
http://www.habbohotel.co.uk/habbo/en/

This is quite a popular animated chat room.