Well, one reason could be Sasuke's continued obsession with his revenge. He might be so focused on it that he ignores Sakura's feelings and concerns, which could lead to a fight. For example, if Sakura tries to persuade him to let go of his revenge and he refuses, it could spark an argument.
Maybe it's about Sasuke's secretive nature. He often keeps things to himself, and in a fanfic, Sakura could get tired of it. She might feel that he doesn't trust her enough to share his problems. So, she confronts him about it, and that turns into a fight.
Another possibility is jealousy. In a fanfic, there could be other characters who show an interest in either Sasuke or Sakura. If Sasuke gets jealous of someone getting too close to Sakura, or vice versa, it can easily lead to a fight between them.
Maybe in the fanfic, Sasuke has done something to betray Sakura's trust. For example, he might have left the village again without considering her feelings.
It could be because of a dark power corrupting Sasuke. Maybe an ancient evil force takes over him and makes him see Sakura as an enemy. In some fanfics, this kind of external influence can completely change a character's actions. Also, it might be related to Sasuke's pursuit of power. If he believes Sakura is standing in the way of his ultimate power - seeking goal, he could be driven to kill her.
One possible reason could be that some fans might want to explore different character dynamics. In many fanfics, Sakura is often paired with Sasuke. But if the story focuses on Sakura avoiding Sasuke, it could be to break the typical mold. Maybe the writer wants to show Sakura's independence and growth outside of her relationship with Sasuke. For example, she could be on a journey of self - discovery and Sasuke represents a past she wants to move away from.
In a fanfic, Sasuke's association with dangerous elements might make Sakura scared. For instance, if he teams up with a group of rogue ninjas. Also, his emotional state could play a role. If he is consumed by revenge and his anger is constantly boiling over, Sakura, who cares about him, might be scared of his self - destructive tendencies. Additionally, if Sasuke were to develop some sort of mind - control ability that he struggles to control, Sakura would surely be frightened.
Maybe to explore different character arcs. The writer might want to see how Sakura would develop as a character if she strayed from her loyalty to Sasuke. It could also be a way to introduce new characters or relationships into the Naruto fanfic world.
One possible reason could be Sasuke's focus on his revenge. His thirst for vengeance against Itachi might make him blind to Sakura's feelings and neglect her as he is constantly preoccupied with getting stronger to achieve his goal of revenge.
One possible reason could be that Sakura has finally grown tired of Sasuke's aloof and cold behavior towards her. She might have realized that she has been constantly chasing after him while getting little in return, so she decides to ignore him as a form of self - preservation.
Perhaps they leave because they feel restricted in the village. They want to explore the world and grow as ninjas outside the boundaries and rules of the village. They might also want to learn from different ninja cultures and techniques that are not available in their own village.
Sasuke might believe that he's a danger to Sakura. Given his associations with dangerous elements and his own dark powers, he may feel that getting close to Sakura would put her in harm's way. Therefore, he rejects her to protect her. This is a common theme in many fanfics where Sasuke is portrayed as having a strong sense of self - sacrifice when it comes to Sakura.
One reason could be jealousy. If in the fanfic there's another character who Sasuke is close to, Sakura might be rude out of envy. Another reason could be that she feels betrayed by Sasuke in some way, like if he made a decision without considering her feelings. Also, it could be part of her own character development, starting off rude to show her flaws and then changing later.