Well, in fanfictions, often the plot takes a different turn. Naruto might hide from his family because there could be some sort of misunderstanding. For example, there could be false rumors about him being involved in something bad. And he might think that his family would believe those rumors without giving him a chance to explain. So, to avoid the immediate confrontation and judgment, he chooses to hide.
Naruto might end up feeling extremely lonely. His family has always been an important part of his life. Without their support and presence, he could become depressed or start making even more reckless decisions. Also, if his family tries to search for him, they could get into danger themselves in the process, which would be an unintended consequence of his hiding.
Naruto could be afraid of being seen as weak. In the ninja world, weakness is often frowned upon. He has a reputation to uphold, especially in front of his family. He doesn't want them to think that he can't handle his problems, so he hides his pain. He also might be trying to be strong for his family. If they see him in pain, it could affect their morale. So, he pretends that everything is okay to keep their spirits high and to give them the impression that they can rely on him no matter what.
Perhaps he has hidden the full extent of his Sage Mode abilities. He might be able to stay in Sage Mode for a much longer time than he has shown so far, or have some other undisclosed powers within that mode, like the ability to sense enemies from much greater distances.
In many fanfictions, Naruto might hide his pain by putting on a big smile. He's always been the type to try and be positive in the face of his friends. He could also distract others with his goofy antics, like making a joke or doing something silly to shift the focus away from his inner turmoil.
He might use simple language on purpose. For example, he could pretend to be confused about things he actually understands well. He may also make silly mistakes intentionally to give the impression that he's not as smart as he really is.
Sasuke could be one of the people who help him. Despite their past differences, they have a deep bond. Sasuke has his own skills and knowledge that could be useful in Naruto's situation, like his Sharingan abilities which could help in detecting enemies while Naruto is hiding his face.
He might use some special jutsu or seals. For example, he could use a sealing technique to suppress the signs of his bloodline. This way, others wouldn't be able to detect it through normal means like sensing chakra.
It could also be part of a strategic move. By hiding his bloodline, he can surprise his opponents in battle. They would underestimate him based on not knowing about his true powers, and then he could turn the tables on them when the time is right. This gives him an edge in combat situations.