I've noticed how much my scalp itches, & the rest of my skin itches, only in the last 5-10 years. And I live in a small town, pop.200, which only started chlorinating the water in recent years,considering I'm 48 & lived here my whole life. There have been times I can smell the chlorine in the sink or shower when it's running. Started to occur to me the last few years this chlorine crap is unhealthy, we were better off years in the old days before they started doing this. People in India & other countries put up with GI problems because of bacteria in their water,but at least they aren't being poisoned to death by the government.
If your not happy with the water then install a well. Chlorine is used to kill pathogens in the water. Seems to me you just have really really sensitive skin... so if you were to go to a country that doesn't have water quality standards as high as they do in the US you'll have issues there too