One possible scenario is that Naruto senses Sakura is in trouble through his ninja instincts. He quickly rushes to the scene and uses his brute strength to fend off the bullies. Just like how he fought against stronger opponents in the original series, he wouldn't hesitate to protect Sakura.
It could be that Naruto has been secretly following Sakura to keep her safe. When the bullies approach, he reveals himself and challenges them to a fair fight. He tells them that if they win, they can continue, but he is confident in his skills and knows he can defeat them, thus saving Sakura from the bully situation.
In some fanfictions, Naruto might save a civilian who has been poisoned. He would use his knowledge of herbs and medical ninjutsu to find an antidote and save the civilian's life. There could also be a situation where a civilian is trapped in a burning building. Naruto would brave the fire, using his chakra to protect himself, and carry the civilian to safety. And sometimes, a civilian is being mind - controlled by an enemy, and Naruto has to break the control and save the civilian.
One scenario could be that Naruto is poisoned during a battle. Jiraiya, with his vast knowledge of herbs and medicine, finds an antidote in time to save Naruto. Another possibility is that Naruto is trapped in a genjutsu created by a powerful enemy, and Jiraiya uses his own genjutsu - breaking skills to free Naruto.
Maybe in the fanfiction, the situation is more complex. Hinata is being held hostage by the Cloud in a heavily guarded location. Naruto infiltrates the place secretly. He uses his transformation jutsu to disguise himself as a Cloud ninja at first to get closer. Then, he reveals himself and uses his Sage Mode to take on all the guards. He finally rescues Hinata and they escape together.
One possible scenario could be Naruto being forced into a situation where he has to seem to betray Sakura to protect her from a greater danger. For example, he might be blackmailed by an enemy into staying away from her or acting against her interests.
In some fanfictions, Naruto might punish Sakura for betraying their friendship. For example, he could distance himself from her, making her realize how her actions have hurt him.
One possible scenario is that Naruto arrives in the Fairy Tail world just as a powerful dark guild is about to destroy Fairy Tail. He uses his speed and combat skills to take on the leaders of the dark guild and defeats them. Another scenario could be that Fairy Tail is facing a magical disaster, like a powerful spell gone wrong. Naruto, with his ability to adapt quickly, figures out a way to counter the spell using his own ninja powers.
One possible scenario is that Hinata and her team are on a mission in an unknown territory. They encounter a group of rogue ninjas. Hinata tries to defend her team but is outnumbered. Just as they are about to be defeated, Naruto shows up, having followed Hinata's chakra signature. He quickly takes down the rogues with his speed and strength.
One scenario could be a competition gone wrong. For instance, if there's a ninja skills competition between Ino and Sakura and Sakura cheats or loses her temper when losing to Ino. She might then try to attack Ino, and Naruto would save Ino. Another could be a love triangle - related situation. If Sakura thinks Ino is stealing Naruto's attention, she could lash out at Ino, only for Naruto to intervene.
Perhaps Naruto has some sort of premonition about Sasuke's danger. He follows his gut feeling and finds Sasuke trapped by Haku. He then uses his Uzumaki bloodline power, something that might be enhanced in fanfiction, to disrupt Haku's chakra flow and save Sasuke.
One possible scenario is that Naruto arrives at the Tower of Heaven just in time. Erza could be on the verge of being defeated by a powerful villain. Naruto uses his chakra to boost his speed and punches the villain away. He then quickly takes Erza and flees the tower. Another scenario could be that Naruto has to solve some puzzles or overcome traps in the tower to reach Erza. Once he does, he uses his chakra cloak to protect them both from any incoming attacks as they escape.