I think that depends on how you were raised/your view of the world. I'd imagine that people who grew up with people who accept the ideas of psychics, crystals, empaths, etc, it feels as though this is perfectly natural. I grew up in a religious family, specifically in a Christian denomination that doesn't accept this sort of thing. However, my mom noticed some weird things about me, specifically, waking up in the middle of night to go comfort when I knew she was very upset even as a toddler and getting sick with ailments that were not contagious when people around me had those symptoms. As a nurse, she'd seen that sort of thing happen enough that she couldn't really deny it. She had back problems, and I naturally knew where I had to press to release the knots in her back (before I had any idea what trigger points were or had any knowledge that something like acupuncture existed) and I even knew where the knots were without needing to feel her back. I remember her asking me lots of questions about how I did that; I'd answer truthfully that I just knew what to do, didn't everybody? The name Empath was never spoken and the "weirdness" I had was acknowledge as something weird, but it was also something that was not discussed. She's the only person in my family that I ever mention any of this empath thing to and it makes her really uncomfortable when I do. The whole point of that was to say, for me, some of this seems natural and not crazy, but it also feels like a great big secret, a piece of myself that I hide from the world. I've tried to lock that piece of myself away, and it generally doesn't turn out well so I still act authentically but only to the point of coming off as sweet person rather than someone who is picking up on their emotions or that their upper right shoulder is bothering them. I'm pretty new to this group, but having a group that accepts that what you experience is real helps make everything feel more real and right.
Also, if you work through the things Trevor Lewis posted and you feel considerably better, there is some evidence to support that this is a real part of you. Technically, as a scientist, I have to say it could still be placebo effect, but there is no harm in trying.
I think another thing that helps with not feeling crazy is looking into other cultures. Plenty of cultures acknowledge energies, spirit worlds, spirits helping people, etc. Just because you may have been raised in a culture that sees this sort of thing as hogwash (as assume you were because you feel like you might be crazy), doesn't make your culture right about this sort of thing. I mean, in many commutative cultures, cultures in which the well-being of the group takes precedence over the need of the individual, this sort of "gift" can be readily accepted and seen as a great thing rather than a burden.
Sorry this is kind of long and meandering, but in short, being aware of your own biases towards how you are viewing yourself, working through the reading materials, opening yourself to new viewpoints, and participating in this group (or another like it) can help you feel a bit more normal, because what is "normal" anyways
Also, when I've felt similar to how I think you are feeling, I needed a big accepting bear hug. So, even though it isn't the same thing virtually, I am sending you a virtual big bear hug of acceptance. You'll be okay.