One possible reason could be that he discovers a dark secret about his family. Maybe he finds out that his parents were involved in something dangerous or unethical in the wizarding world, and he can't bear to be associated with it, so he runs away.
It could also be that he is being targeted by some dark forces because of his parents. He runs away to protect his parents from having to constantly protect him. He believes that by leaving, he can draw the danger away from them and figure out a way to deal with it on his own.
He could go to Hogwarts early. Since he has some connection to the school already through his parents, he might think that the school could be a safe haven. There are also his friends' families there, like the Weasleys, who might take him in.
He could be running away because of false accusations within the family. Just like how he was wrongly judged by the village in the original story, if his family wrongly blames him for something he didn't do, like stealing a family heirloom or causing harm to a family member accidentally, he might choose to run away rather than face their unjust wrath.
Maybe he discovers some dark secrets about his family. For example, if his parents were involved in something he considered unethical in the fanfic's plot. It could be that they were part of a group with a hidden and evil agenda.
One reason could be fear. Snape has always been rather intimidating to Harry. Snape's cold demeanor and his past interactions with Harry's father might make Harry feel constantly on edge around him. So, running away could be an instinctual response to this fear.
One possible reason could be that in the fanfic, the author might twist the story to make it seem like his parents' actions had unforeseen consequences that affected Harry negatively. For example, their fight against Voldemort led to Harry being an orphan and having to bear a great burden from a young age.
It might be that his family is overprotective. Naruto is a very independent character. If his family in the fanfiction is constantly trying to hold him back from taking risks or going on adventures, he may feel suffocated. Running away would be his way of breaking free and exploring the world on his own terms.
Maybe he wants more independence. In many fanfics, characters like Harry might feel restricted at home and long for the freedom to make their own choices, so that could be a reason for him to move out.
In fanfic, one reason could be that he might feel constantly in danger at Hogwarts. With all the dark forces targeting him, he could think running away is the only way to keep his friends safe. For example, if Voldemort's followers keep infiltrating the school, Harry might feel responsible and decide to leave to draw the danger away from others.
One possible reason could be that he might feel abandoned. Growing up without them, he could resent the fact that they left him an orphan. He might think that if they were really so great, they would have found a way to survive and be there for him.
Maybe Harry feels a sense of guilt. In the context of the war, he might think that he's failed in some way by getting into this unexpected situation. He could believe that his new condition might put others at risk or that he's no longer the hero everyone expects him to be. So, to protect others or to avoid the shame, he decides to run away.