In the fanfic, it might be that Naruto has new responsibilities in the village that are more important than chasing Sasuke. For example, he could be needed to deal with new threats to Konoha, so he has to let go of his personal mission to bring Sasuke back.
Well, it could start with Naruto having an internal struggle. He might have a moment of realization that chasing Sasuke has cost him a lot and isn't really getting him anywhere. So he decides to focus on other things, like training new ninja in the village.
In the fanfiction, Naruto could stop chasing Sasuke because he has a moment of self - realization. He might understand that his own identity and purpose in life are not solely defined by bringing Sasuke back. He could find new goals and dreams that don't involve Sasuke. Also, there could be a major event in the ninja world that makes Naruto shift his focus away from Sasuke, like a new and powerful enemy that threatens the entire village or region.
One possible reason could be that Naruto has faced too many betrayals. Maybe his friends turned on him in the fanfic, which made him lose faith and stop caring.
One possible reason could be that the author wants to focus on other characters. Maybe they feel that Naruto's story has been told enough in the fanfic and it's time to give other characters the spotlight.
One reason is that Naruto has the Nine - Tails' chakra. In fanfic, this power can be further enhanced or manipulated in ways that make him far more powerful than Sasuke. Another reason could be his more outgoing personality. He might be able to form alliances more easily, leading to him having more resources and power - ups at his disposal.
One reason could be to create a lot of angst. If Sasuke cheats, it would deeply hurt Naruto and cause a rift in their relationship, which could lead to a lot of emotional turmoil for both characters. This would make for a very dramatic story.
One possible reason could be that it goes against the original dynamic between Naruto and Sasuke. In the original story, their relationship is based on a deep male - male bond of rivalry and friendship. Changing Sasuke to a female might disrupt that established connection.
One possible reason could be a difference in their goals. Sasuke has always been on a journey of revenge and self - discovery. If in the fanfic, Naruto's path starts to conflict with Sasuke's in a way that Sasuke can't tolerate, he might leave. For example, if Naruto decides to fully side with the village elders whom Sasuke distrusts, Sasuke could see this as a betrayal of their friendship and leave. Another reason could be an external threat that forces Sasuke to leave to protect Naruto. Maybe there's a powerful enemy that specifically targets Sasuke and he doesn't want Naruto to get caught in the crossfire, so he leaves without explanation to keep Naruto safe.
A moment of desperation. For example, if one of them is about to die in the alley due to an injury or a curse. The other, in a rush of emotion, kisses them as a way to convey all the unsaid things, all the love and friendship that has been there all along.
Another possibility is that Naruto was trying to protect someone else from Sasuke. Suppose Sasuke was on a path that would endanger another important character in the story, like Sakura. Naruto, in his eagerness to safeguard Sakura, might have accidentally hurt Sasuke while trying to stop him from taking any harmful actions.