Nnno, the conceit, at least since the late '80s, is that he has a bulletproof costume. (Frank Miller established that the bright yellow oval was a target so bad guys would aim for the most heavily armored part.) Also, part of the reason for the cape is that it obscures his silhouette so it's harder for the bad guys to hit his body.
In Batman '66, Batman and Robin had a bulletproof Bat-Shield, like a clear, unfolding version of a police riot shield.
Of course, no mere mortal could move fast enough to literally dodge a bullet. But it's hard for a shooter to hit a moving target, so as long as Batman and Robin didn't stand still but came in swinging (in either sense of the word), they would be difficult to hit. And they often used batarangs to knock guns out of bad guys' hands before the bad guys even saw them.
I hate myself for doing this, since we both agreed already that Batman is not a realistic concept, but fuck it, let's open that other can of worms.
Yes, the comics (and at least some of the movies from the 1989 version on) have incorporated the bulletproof costume, and the comics at least named kevlar as the material used. And, yeah, kevlar is the material used in protective clothing like bulletproof vests, so fair enough, kevlar would block bullets - though they still would have an impact, like in the 1989 film where they still punch Batman to the ground, and not making them totally ineffective like in most comics or, famously, Matt Reeves' The Batman - but it is inflexible AF. Why do you think they stopped at bulletproof vests and don't make a whole suits for law enforcement and/or the military out of the stuff? People wouldn't be able to move around. And Batman moves around a lot more than your basic police officer.
That was even adressed in The Dark Knight when Lucius Fox designed the new suit for Batman, where he points out that, in order to make the suit more flexible, he had to subtract from the protective armor.
So, realistically, Batman could wear a protective suit, but would be hit by a hell of a lot more bullets, or he could, you know, dodge bullets. Not both, and both ways would eventually lead to a bullet getting through.
People make fun of how stiff Michael Keaton's Batman was, but at least this is consistent with the suit being lined with kevlar. Which means, in this regard, Tim Burton's Batman is more realistic than other, "more grounded" versions.
Not to mention that kevlar is also heavy, which adds to the whole exhaustion thing.
And even the comics and most movies weren't stupid enough to claim the mask being bulletproof, meaning of the multitude of bullets being shot at him, it would also be just a matter of time until one would hit him in the head.
Except for the DCEU, which was that stupid and specifically showed Batman's mask to be bulletproof, while trying to be "grounded and realistic". *sigh*