Being a relative of a cop in the jurisdiction in which you ran someone over will prevent you from ever being charged with it the US (they'll probably give you some slap on the wrist charge though).
Being good friends with the right folks at city hall will make sure your company wins the contract to do the landscaping at the city park.
Knowing the right people will generally get any permit you need expedited or at the very least it won't be delayed.
If special treatment is corruption then corruption is very much alive and well in the US. The difference is that the image of corruption is corruption is much less tolerable. If the details of the story of you running over someone makes the news or someone points out the conflict of interest of the person awarding the contract the corruption won't happen in that case. "I'm sorry I can't do that" is a much less common phrase than "I'll put in a good word for you".
Corruption in US (federal, state and local) is a lot like the restaurant that only cleans up its act for the health inspector. Government is the restaurant and publicity is the inspector. When there's a spotlight on the problem everyone plays by the rules. When nobody's looking it's business as usual.
Acts of obvious corruption (trading favors, bribes, etc) is rarer in the US than some other countries but leveraging relationships for special treatment is just as common as anywhere else, we just keep it under the table.
This isn't an attack on government but government is certainly not the consistent and impartial organization that some people seem to think it is (though I think that's a good goal to have).
Edit: I find it odd/interesting that whenever I say something that isn't positive about government it gets up-voted a couple times then down-voted a couple times. I really should write a script to track it because I have a hunch that the timing follows a particular pattern.