It could also be a misunderstanding. Perhaps Karin is misjudged by Sakura in the fanfic. Maybe Karin did something that was misinterpreted as malicious towards Sakura or someone she cares about, leading to the hatred.
Often, it starts with the establishment of their dislike. Then, there could be a series of confrontations. For instance, they might argue or compete in some way, like in a ninja mission where they constantly try to outdo each other.
In some cases, Karin can be more popular. Her backstory of being associated with Orochimaru and her survival instincts make her an appealing character. Some fans might be drawn to the idea of her as an underdog in the love triangle with Sasuke and Sakura. Also, her sensing abilities and her role in certain plot points in the original series can be explored more deeply in fanfics, which might make her more interesting to some readers than Sakura.
I'm not sure specifically as it depends on the particular fanfic. It could be about a situation where Karin is shown bullying Sakura within the context of the fan - created story.
Maybe Karin has some personal reasons. For example, she might not like Sakura's character in the original work, so she also dislikes fanfic related to Sakura.
Often, Sakura is depicted as kind - hearted. She is the one who is more likely to see the good in people. Karin, on the other hand, might be more headstrong. For example, when they face a difficult situation, Sakura will try to find a peaceful solution first, while Karin is ready to jump into action right away.
Love triangles are a very common theme. Sasuke is often at the center of the attention of both Sakura and Karin, which creates a lot of tension and drama.
Well, some specialized anime or manga fan communities might also be a good place to search for sakura karin fanfic. There could be forums or groups dedicated to the particular series that Sakura and Karin are from, and fans often share their fanfic creations within those communities. You could also try searching on social media platforms using relevant hashtags, but the content might be more scattered there.