Do I draw a new hand if I start my turn without any card?
No, you do not draw a new hand when yours runs out. Your friends are playing with a house rule. It might be an optional rule in some expansion's handbook, but I'm not aware of such an optional rule.

The only time you draw a new hand is when you start a new character: at the beginning of the game, or after you die.

You are not expected to always have cards in your hand in Munchkin. In fact, it's highly likely that you won't have any cards in your hand a while into the game, and that's fine!