I don't want to get into why Aegon bullied Aemond at all, but I do want to touch upon the question of how he stopped. Because he did.
When the episode about the Battle of Rooks' Rest was released, and Aemond burned Aegon, there were opinions like, you know: Aegon has only himself to blame, he had been bullying Aemond, he had been trampling Aemond into the dirt for so many years, since childhood, and now, look, he's doing it again in that brothel scene in episode three, he never stopped, so Aemond couldn't stand it anymore and just went and took revenge. Something of the sort.
But the thing is... that's not true. Not exactly. In season one, before the family dinner scene where it all turns to shit, we see the brothers talking, and if you turn up the volume and filter it out the noise — I think someone's done that before — and listen to their conversation, straight away it's obvious that they have a kinda normal relationship. Moreover, they share their thoughts with each other. This means they had no serious confrontations for quite a long time. You don't share your thoughts with someone you're not on good terms with. When the fight breaks out, Aegon jumps up from his chair being ready to help his brother. Although, let's be honest, Aemond isn't a character who needs to be saved... My point is that Aegon hasn't bullied Aemond for some years, and their relationship is much better at the time than it was when they were children. So Aegon has stopped his cruel jokes.
I think one of the reasons why Aegon chose Aemond as the butt for ridicule was the fact Aemond couldn't hit him back at that time. He was young and weak. There was nothing Aemond could do. It's easy to poke fun at someone who cannot punch in the face in response. The one who's stronger physically yet rat bastard morally is often violently inclined against those who can't defend themselves. This is precisely why some people, mostly men — I can't imagine why — hurt women, children, and captives. Sad but true.
However, everything has changed the night when Aemond got Vhagar and lost an eye. Since then, he is no longer a punching bag. He had been through the fighting against a large, winged, igniferous beast and won. Vulnerability and helplessness? This isn't about Aemond any longer. The same night Aemond said, in front of everyone, that speculations about Luke and Jace's parentage were whispered to him by Aegon. But we know Aegon didn't do that. It's borne out by his reaction to charges leveled against him: he seemed so surprised. He was not denying Aemond's statement nevertheless. Because Aemond had stopped being just a little boy who cannot fight back. Aemond was younger, always would be, he was still weaker physically, but he has become a formidable enemy. He showed his brother, showed them all, what he was made of. He showed his inner fire, his power, his fortitude and strength of mind. His victory over Vhagar was a victory over his own weaknesses and fears. From then onwards, Aemond is no kind of victim or butt of jokes. So yeah, Aegon has stopped acting like a piece of cruel shit towards him, because he would be hauled over the coals otherwise, and he knew it.
Wait, what about that brothel scene? The one where they both are adults?
It's hard to say for sure, whether Aegon held Aemond responsible for his son being murdered. Maybe he did. Maybe he didn't. But Aegon was drunk and crushed down by grief, that's for sure. He saw a flash of vulnerability in Aemond's eyes — sorry, an eye — for the first time in years. I think a strong instinct to hurt came up inside Aegon. It was as if his inner spring came loose, no wonder it turned to mocking that punched Aemond in the face. He did not think too much about what he was doing. All he cared about was to blot out his own pain, and some people can end their suffering only by causing more suffering to others. It was what it was. He made a big mistake of allowing a laugh at grown, strong, vindictive, relentless Aemond. If he weighed up the pros and cons, he wouldn't do that, he would know better that he wouldn't get away scot-free. Or maybe he had a devout belief that a king could get away with anything, as an option.
— Tea & Rum, The Cracked and the Broken.
