Am I wrong in accepting bullying as a normal human trait? 
Yes you are wrong, because civilised society is based on civility, if we lived in a law of the jungle Anarchist society, bullying would be normal, but when humans are constrained by laws, rules and regulations, were a premium is put on civilised behaviour, for those who suffer from some sort of personality defect and don't care about themselves, their reputation or consequences for their actions, it is unfair for these people to single out others and push the boundaries. If we pass a law that states bullies can be beat the fuck down to the point of near death, then it wouldn't be so bad, and bullying wouldn't happen as often, but that type of law would only move civilised society one step closer to the law of the jungle anarchist system. Those who are prepared to act in accordance with civilised society shouldn't be negatively impacted by those that aren't. Personally I would prefer a system that allows the bullied to be able to fight back and freely inflict suffering on their tormentors with little to no penalty, too much freedom is given to those wanting to cause trouble, if the law turned a blind eye, decent people could have this sorted in a heartbeat.
Society should always remain focused on determining the person or group that started the altercation, civilised society isn't made by laws, rules or regulations, it is made because the majority of people are civilised, bullies generally lack civility, in my experience the best way to deal with them is too beat them down. In my adult life, I have had the pleasure of seriously injuring a bully, and it was a very satisfying moment indeed.
A word for the bullies, make sure the person you bully doesn't have the capacity to beat you the fuck down.
