Seeing how the setting was mainly East Asian based, I thought the rule was First born son would become heir, regardless of how many elder sisters the boy had, then the second, third etc. son, then the eldest daughter etc.
But it kinda makes sense too in another way. I mean, Azula might be acting this way just because she wants to prove herself to be better than Zuko.
(But still, hard to imagine some 14 year old to act like this XD)