One reason could be that Naruto might feel he has outgrown Jiraiya's teachings in the fanfiction. He may want to explore his own path independently. Another possibility is that Naruto gets involved in a mission or a situation where he has to part ways with Jiraiya for the greater good. For example, if there is a threat in a far - off land that only Naruto can handle, and Jiraiya has other commitments elsewhere.
In some fanfictions, Naruto could leave Jiraiya due to a misunderstanding. Maybe Naruto overhears something Jiraiya says that he misinterprets. For instance, if Jiraiya is talking to someone about Naruto's progress and Naruto mishears it as Jiraiya doubting his abilities. This could lead to Naruto feeling hurt and deciding to leave to prove himself on his own.
Perhaps in the fanfiction, Naruto leaves Jiraiya because he discovers a new power source or a new group of allies that he believes can help him achieve his goals better than Jiraiya can. He might be lured by the promise of faster growth or more effective training methods. It could also be that Naruto is influenced by other characters who convince him that it's time to move on from Jiraiya's tutelage.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he wants to explore his own path independently and test his skills without Jiraiya's constant guidance.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he has reached a level in his training where he believes he needs new experiences and challenges that Jiraiya can no longer provide.
One reason could be to seek more power. In some fanfictions, Naruto might feel that the Leaf doesn't offer enough resources or training for him to reach his full potential. So he decides to leave to find a place where he can grow stronger.
One possible reason could be a difference in their goals. Naruto is from the Hidden Leaf Village with a mission to protect it and promote peace there. Rias, on the other hand, has her own responsibilities in the Underworld. Their conflicting duties might lead to Naruto leaving.
One possible reason could be that in this fanfiction, Jiraiya might be preoccupied with other important matters. Maybe he has a secret mission that demands his full attention and he simply doesn't have the time to train Naruto. Another reason could be that there is some sort of conflict or misunderstanding between them that prevents Jiraiya from taking Naruto under his wing. For example, Naruto might have unknowingly offended Jiraiya in some way, or there could be a prophecy or external force that is keeping them apart for the time being.
One possible reason could be that Naruto might feel Jiraiya is being too overbearing in his training methods. In some fanfictions, Jiraiya might push Naruto too hard without considering his emotional state, leading to Naruto snapping.
Perhaps Jiraiya has some personal issues or inner conflicts. He might be dealing with his own past traumas or demons that prevent him from taking on the role of Naruto's trainer. For example, he could be haunted by the failures in his previous relationships or battles, and he feels unfit to train Naruto until he sorts out his own problems.
One reason could be to gain more power. In many fanfictions, Naruto realizes that the threats he faces are far greater than what he can handle with his current skills in Konoha. So, he decides to leave and train in a more secluded and challenging place to enhance his abilities.
Perhaps Naruto has some unresolved issues from his past that he doesn't want to burden his family with. He may think that leaving is the best way to deal with these issues on his own, without putting his family in the middle of his personal turmoil. He might go off on a solo adventure to find himself and come to terms with his inner demons.
One reason could be to seek new challenges. The elemental countries might have become too familiar, and he wants to explore uncharted regions to test his skills further. Another could be to escape from the fame and expectations in the elemental countries. He might feel suffocated by the constant attention and the high standards set for him.