They might have adventures in training Naruto to control the Nine - Tails' chakra better. Jiraiya could take Naruto to different secluded places where he can practice without distractions. Maybe they encounter strange and powerful beasts along the way, which Naruto has to learn to fight using his new skills.
Naruto's character develops through the training he receives from Jiraiya. He becomes more disciplined. For example, he learns to wake up early every day for training without complaining like he used to in Konoha. His ninja skills also improve, which boosts his confidence.
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.
Konoha would initially be in shock. Naruto is a hero to many in the village, and his sudden departure with Itachi, who was once considered a traitor, would cause a lot of unrest. The villagers might start to question the leadership of the village, wondering why Naruto left and if there was something they could have done to prevent it.
Jiraiya's reaction might be complex. He might be in awe of Naruto's new power but also sad about his departure. He could see it as a natural progression in Naruto's journey, yet still feel a pang of loneliness. He might start to reflect on his own life and training methods, wondering if he could have done something differently to keep Naruto with him. At the same time, he would be eager to see what Naruto will achieve in the world with his newfound abilities.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
One possible reason could be to seek stronger opponents outside of Konoha. Naruto might feel that he has outgrown the training opportunities in the village and wants to face greater challenges.
This type of fanfiction concept is really out of the ordinary. Naruto's character is built on his values of friendship, loyalty, and his dream of becoming Hokage. Killing Sasuke would mean betraying all of that. However, in a fanfiction, the author might be exploring the 'what - if' scenario. For example, what if Sasuke's actions against Konoha were so heinous that Naruto felt he had no choice? But still, it's a far - fetched and very different view from the original Naruto we know.
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.
It could lead to political unrest in Konoha. There might be those who blame the village leadership for Naruto's departure. This could lead to power struggles as different factions try to take control and change the way the village is run. Some might demand more transparency and better treatment of their shinobi to prevent such a thing from happening again.