There are clean, western-style toilets in shopping centres, malls, department stores and most restaurants in Bangkok. Sometimes in a mall you have to pay 2 or 3 baht. Off the beaten track, you may encounter Asian-style squat toilets.

In general, Thai plumbing is not engineered to deal with toilet paper. You're supposed to wash yourself with water (from a water jet attached to the toilet, or a standing barrel of water) and then use the paper to dry yourself off and throw in the basket, not in the toilet bowl.

It's a good idea to carry some wet tissues to wash your hands. In simple places, hand-washing facilities are often less than adequate.

Some shopping malls will have a choice of Asian-style or western style. Toilet paper (note the comments above) is often dispensed from a machine for 2 baht. Gas (petrol stations) often have clean facilities.

On top of shopping malls, I often use toilet facilities at decent hotels when I am getting around Bangkok etc. They are all western toilets as far as Bangkok is concerned. Asking a receptionist if you could use their bathroom would be very nice.