I'm going to be picky about this. To be a superhero you need a superpower. Batman is a hero, or rather an antihero, but not a superhero. Though, as I understand it, recently they've started changing that so any heroes with no powers are actually meta-humans, or some such nonsense. Which basically means they're the next stage in human evolution. Faster, stronger, smarter, better reflexes. That sort of thing. But I'm a bit of a purist and I'd rather think of Batman as a good honest psychopath.
So the X-men end up in a bit of a grey area between superhero and meta-human. It basically depends upon personal preference. And there was that thing on Sky one, that had X in the name of it, but I've completely forgotten the name of it. Had the woman with the cat eyes who could leap around a lot, the girl with the bad hair who messed with minds, the chap who could turn solid like rock or vapid like the plot, and the baddie was Andi Warhol running some company called Gencorp. They all called themselves New Mutants, or metahumans.
TV is steering away from the superhero concept a lot these days, as it involkes images of burly men in brightly coloured lycra that just aren't as appealing as they were in the 50s.
And then you've got the Green Lantern. What's he? He's got all these super powers when he's wearing his ring, but when you take it away he's a bit of a wuss.
So basically, I think my point is this. In order to win your argument, you can claim that although these heros with no powers aren't supernatural, they do possessed enhanced abilities with regard to normal humans, making them meta- or super-human. So while they may not be superheros in the classic sense, they're still an evolutionary step above us, so they can kick our arses so they might as well be superheros as far as we're concerned.
And yes. I do watch far too many cartoons.