You can definitely blow out tires from overheating. It's a bit rare on a single bike with aluminum rims, and not carrying a load.

If the rims get really hot, you'll smell and hear the pads off gassing, at which point you need to stop.

The key is to not brake constantly. Brake firmly before a turn, then let your speed build, until the next turn, and then brake firmly again. This allows the pads and rims to cool a bit.

What you don't want to do is "ride the brakes" which it sounds like you're doing.