In some fanfictions, external forces could be the cause. There could be a threat to Katara's safety, and Aang leaves her as a decoy to draw the danger away from her. He values her life more than their being together and decides to sacrifice their relationship to keep her safe. Another aspect could be that the Avatar's duties require him to go to a dangerous place where he doesn't want to put Katara at risk, so he leaves.
Maybe in the fanfiction, Aang leaves Katara because of a misunderstanding. For example, he could misinterpret something Katara said or did regarding their relationship. This misunderstanding could lead him to believe that it's better for both of them if he goes away. He might think that he is protecting Katara in some way by leaving.
Perhaps Aang has a vision of the future where Katara is happier without him. So, in the fanfiction, he decides to respect that future and let her go.
Katara could be heartbroken. She might initially be in denial, not believing that Aang has actually left. Then she could become angry, feeling betrayed by Aang's sudden departure.
The writer might be trying to add some drama to the story. By making Aang hate Katara, it creates a conflict that can drive the plot forward. For instance, their friends might have to work hard to resolve this hatred and restore their friendship.
In some fanfictions, it might be part of a plot to create tension. The writer wants to show that their relationship isn't all smooth sailing. So, Katara hurting Aang can be a device to make the story more interesting. It could also be related to different character arcs, like Katara trying to become more independent and in the process, her actions accidentally harm Aang.
It could be that the fanfiction writers wanted to explore a more complex and darker side of relationships. They might have written scenarios where Aang hates Katara due to jealousy, perhaps over her friendship with another character. This would add an interesting twist to the original relationship dynamics between them.
It could also be that Katara was being overly critical of Aang's progress in learning new skills. Aang might feel frustrated and angry if Katara is constantly pointing out his flaws instead of being more encouraging, especially if he's already trying his best.
One possible reason could be their different approaches to life post - war. Aang is the Avatar and has a whole world to look after, while Katara has her own tribe and personal goals. Their busy schedules might pull them apart. For example, Aang might be constantly traveling to different nations for diplomatic and Avatar - related duties, leaving Katara feeling lonely. Also, they might have different ideas on how to handle future threats. Aang may prefer a more pacifist approach which Katara might not always agree with, especially if it endangers the people she loves.
Perhaps Katara falls in love with someone else. In a fanfiction scenario, she might meet a new character who she feels a stronger connection with. This new relationship could cause her to break things off with Aang, shattering his heart in the process.
One possible storyline could be them traveling the world together to help rebuild the nations. They could use Aang's Avatar powers and Katara's waterbending skills to bring water to drought - stricken areas or help with construction using earthbending - like techniques. Another could be them settling down in the Southern Water Tribe and starting a family while also teaching the next generation about the importance of the four nations' unity.
You should check out 'Aang and Katara: Love in the Four Nations'. It takes them on a journey across all the four nations, and during this journey, their relationship blossoms. It also gives great insights into the different cultures of the Avatar world and how Aang and Katara interact with them.