To me, Azula suffers from a combination of mommy/daddy issues, spiraling out into overall trust issues.
Because of what happened with her mother and Ozai, she never felt any sense of love from her mother, and when she finally left, it only confirmed her opinions, making her rely on daddy for emotional stability. But when your father's a power-hungry emperor looking at world domination, his rocker isn't exactly stable.
Combine that with the fact that she was an exceptionally talented firebender and would routinely wreck the mook Fire Army troops, when her dad ordered her to go deal with the Avatar she ran to the only people she could rely upon: Ty Lee, master gymnast and probably one of a handful of people who know about chi blocking at that point in time, and Mai, an equally skilled martial artist and expert with small, pointed objects.
Keep in mind that basically every fight they had was Aang + one of the other benders fighting Azula, while the remaining bender and Sokka duo'ed Ty and Mai, and that was barely successful in most of their encounters, with the Gaang ending up having to retreat after a period of time due to lost ground or needing to advance the plot.
But as Ozai's faith in his daughter grew, her desire to please him (partially due to her emotional issues, partially out of fear) lead that to try and rule over Mai and Ty Lee with fear too (because it's working so well for daddy, it'll work for me too, right?)