One possible plot development could be that Naruto quickly masters basic chakra control under Jiraiya's strict training. This early success might make Naruto overconfident, leading to a challenge where he realizes he still has a lot to learn. For example, he might face a powerful opponent in a sparring match arranged by Jiraiya and struggle initially, teaching him the importance of continuous training.
In the fanfiction, Jiraiya could start training Naruto early in advanced ninjutsu. Naruto might then use this new knowledge to protect his friends in the village from a small - scale threat. This would not only show his growth but also strengthen his bond with his friends as they see him taking on a more protective role. Maybe he uses a new jutsu to save Konohamaru from some bandits.