When you arrive, buy yourself many packs of mini-tissues and never go anywhere without a pack or two on your person. A lot of the restrooms don't have TP in the stalls - instead you toss a few baht coins into a dispenser which gives you a tiny box of a few tissues. Bringing your own is a much better idea, though.

Remarkably I didn't encounter any of the squat-toilets in Bangkok or Phuket, and the only slightly suspicious toilet I came across was at a viewpoint somewhere in Phuket - there was a drum of water and a bucket and you had to do the manual bucket-flush.