I feel as though a lot of people are under the impression that Ozai treated Azula nicely. Yes he did indeed buy her all these nice things but at the end of the day she was pretty much just a pawn to him. Azula may have had her father's respect, but it didn't seem like she had his love
While yes--as far as we know--Ozai never put his hands on Azula, the verbal abuse was apparent. She is afraid of him. Whist we haven't really seen him abuse her verbally I think it was hinted when he yelled at her before declaring himself the phoenix king. I also couldn't imagine he let her down easy behind the scenes after Zuko informed him that Aang wasn't really dead and she lied to him.
She was afraid of him. In that scene (in the comet) when he's yelling at her you can tell that she's afraid. You can see it in her eyes; fear and disappointment. Along side that scene I have this book where Sozin's comet is told from the points of view of the characters. This chapter was from Azula's POV. In it she says something along the lines of "he looks really angry, I've never spoken to my father like this before, is he going to punish me how he punished Zuko?' She's literally wondering if he's going to burn her face off for asking to go with him. A child should fear that his/her father is going to burn his/her face. Even though he respects her (for the most part) Azula still doesn't put it past him to not
Going again back to an above scenario. Why would she lie to her father about herself being the one to kill Aang? Now I may be reading to much into it but; that glory turning to shame thing is probably only a fraction of her motives. I mean really who would be ashamed of Azula for not taking the Avatar down? She's the princess, she took Ba Sing Se. The only person I can think who would give her a hassle for it is Ozai. She gave Zuko credit for killing Aang because she was afraid
of how her father would treat her if he found out Aang was in fact alive. As mentioned above, I can't imagine that he'd go easy on her. It's what happens behind the scenes that counts. I'm kind of surprised that Azula's fate upon Zuko exposing her lie was so overlooked by the fandom.
Which leads me to my next point; you shouldn't have to earn your father's love. It should be unconditional. I think most everything Azula has done was to impress her father and gain his affections from taking Ba Sing Se to going at Aang and perfecting her bending. She even named Omashu for him; the city of new Ozai. I also think she bullied Zuko to impress and be like her father--this is something Zuko always got sympathy points for (doing what he did to get Ozai love). And the sad part is she kept doing so which goes to show that it was working...and moreover that her political success and bending mastery were the only things that got her his love. And it wasn't even really genuine love.
Nope, even then he was pretty much just using Azula. She was like a pawn. Up until the very end her sent her out to do his bidding for him. She sent her to capture her brother and to take Ba Sing Se. Sound familiar? Ozai did the same thing to Zuko in sending him to go get the Avatar for him. Ozai uses both of his children and he does so without regret. And both listened because all they wanted was his approval. And as stated above Azula did everything he asked but at the end of the day he took her idea (to burn everything to the ground) and left her behind. Sure he gave her the firelord title, but he only did so because he knew he was gonna be the all powerful phoenix king. Once that happened, 'firelord' was just an empty title with nothing to it. And he knew it. He just tossed it to Azula to shut her up. And that's not right.
Which leads me to my last point, and this is something that bugs me more than all else. Going once more back to the phoenix king scene; when Azula was standing before him it was so apparent that she was already starting to slip. You can see it in her eyes that she was tired. That something was off. She had faint bags under her eyes and her voice didn't seem as calm. Ozai didn't even seem to notice. How can you look at your own daughter and not realize something was wrong? And you know what, maybe he did notice. And that makes it all the worse because; How can you look at your daughter in such a state and just leave her?! Moreover how could you place the stress of ruling a kingdom (even for that short time before it became a useless title) on someone who's already at her breaking point. If he caught on to her fragile state the fact that he had literally no issue departing...no hesitation at all whatsoever is disgusting. The negligence right there kills me. And by all means negligence is a form of child abuse. Again, this may just be me reading too much into it but; perhaps when Lo and Li come to check on Azula the reason she asks if her father sent them to see how she's doing could be out of wishful thinking. She wants to think her father was concerned. I can also interpret that as proof of one of the above paragraphs; she just wanted her father's love. Nothing she did was good enough for him and he just wanted more and more. Either which way it doesn't paint Ozai in a good light. As he was taking over the world his daughter was losing herself. Losing her mind.
I'm not saying any of this takes away from the fact that Zuko was abused, not at all. However I am saying that Zuko wasn't the only victim of his father's cruelty.