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.
Perhaps Sasuke feels that he is on a different path than Sakura. He may see himself as a lone wolf in his journey, whether it be in terms of power - seeking or self - discovery, and he views Sakura as someone who doesn't really fit into his world at that moment, so he neglects her.
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.
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.
She might be initially heartbroken. Sakura, being in love with Sasuke, would surely be deeply affected when he neglects her. She could become withdrawn herself, losing some of her usual cheerfulness.
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.
Maybe in the fanfic, Sasuke has some personality traits that become more difficult for Sakura to handle over time. For example, his brooding nature might start to feel like emotional unavailability rather than just his usual self. This could lead to her regret.
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.
Some fanfic authors might write it as a form of wish - fulfillment. They might prefer the Naruto - Sakura relationship more than the Sasuke - Sakura one. So, they create a scenario where Sakura turns to Naruto in a romantic way. Additionally, it could be a way to explore the characters' personalities further. For example, how Sasuke would handle being betrayed and how Naruto would deal with being involved in such a situation. It gives a chance to dig deeper into their emotional and moral aspects.
In such a fanfic, one possible reason could be Naruto's feeling of being left out. Maybe he sees Sakura's constant focus on Sasuke and feels neglected in their friendship triangle. Another reason might be jealousy. If Naruto has unrequited feelings for Sakura, seeing her love for Sasuke could turn to hatred over time.