I can't explain it, but Batman needs to be mythologized. In Gotham, that is.
Logically, they know he's just a man, made of skin and bones and blood, like all of them. Those who met him, who had the ill luck to, say he feels like the city itself.
When danger approaches, with the click of a gun or hiss of a knife, it’s not the police people pray to. There's a single name that rivals Gotham‘s, and it’s not them.
So much so, there's whispers of what might happen, about guidelines being stepped. Kinda like how if you're compared to Aphrodite, you might die.
Such as:
If you see a wounded bird, of any kind, take it to a vet, or nurture it back to health. You’ll get robbed otherwise.
Don't wear pearls. Just don't.
Never let your son walk alone at night. Never lay your hands on them, either. The Bat doesn't take kindly to it.