Hard to be a superhero without superpowers. Batman is implausibly well-trained, uses weapons, wears armor, uses every tech trick money can buy, and travels in military grade vehicles. Even then, I think a real-world Batman would have a very short career, if not short life (e.g., Parkour fails, getting shot in the face, police helicopter tracking, GPS tracking, the Batmobile getting towed, someone finally doing a little research into who makes Batman all his stuff, the reality of 6 vs. 1 combat). The most realistic superhero is really a person with superhuman powers. I can see why the cops would lay off Superman. What are you going to do, arrest him? Are you going to catch the flash? Are you going to run up a building after Spider Man? Even so, the anonymity of these characters is terribly implausible -- so you have the problem of where do you sleep? Where do you do eat? What do you do when baddies and governments lean on your loved ones to get leverage? A Superman who lives on the moon in a fortress of solitude and comes down just to put work in is probably the most "realistic" superhero I can imagine.
Even if everyone were part of his extended "family," his allies, his informants, his fellow members in the Justice League or Outsiders (not to mention the plethora of still-living villains who learned his secret)... the law of averages would dictate some minor slip-up by someone along the way that would make his identity very traceable (heck, they even wrote into canon that his second teenage Robin figured out Batman's identity all by himself).
After several seasons of Dexter, too many people also knew who he was. Sure, they kept killing people who knew his secret off, but many remained. There were too many incidents (including official police incidents) to tie him to events that couldn't just go on never being connected by people hunting his serial killer identity. And they had to keep coming up with more far-fetched plot lines whenever someone did begin putting 2 and 2 together.
An aside: I remember as a kid watching the Batman TV show and wondering every episode - why don't they just shoot him and Robin? The villains always set these elaborate death traps (which they'd never stick around for to watch the dynamic duo's ultimate demise and from which the caped crusaders would always escape). I'd always say: what if the Joker just pulled out a gun and shot them both right now? I had a lot to learn about both TV entertainment and the fact that the Joker would never kill Batman - he completes him!