Killing Women

In the first few minutes of The American, George Clooney's latest film (not half as entertaining at the coffee commercials he does with John Malkovich here in Australia), Clooney gets ambushed by a couple of gunmen and once he deals with his attackers he shoots the innocent, unarmed woman he is with in the back of the head so she can't be a witness against him. From that point I had a hard time having any sympathy for Clooney's character. He murders an innocent woman and we are then supposed to care about him and his life after he falls in love with a prostitute. I kinda didn't.
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In the opening scene of The Walking Dead a Deputy Sheriff shoots a little girl in the head because apparently she is a zombie. (Although we don't actually know this at the time). The director hides nothing from us. We get to see the little girl take the bullet in her forehead and die.
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Of course killing women has been a Hollywood staple since the talkies began, but I wonder if some new line has been crossed here? Is there a difference between what these characters did and what Norman Bates does in Psycho? Clooney and Perkins are both playing anti heroes, right? And what about Casey Affleck in the new Killer Inside Me? Can we have any sympathy for him at all? I know I didnt. I have only questions. Not answers. Sorry.