There is a lake in my town where public swimming is allowed, and they test the water regularly and sometimes have to close because of bacteria levels that are too high. Swimming pool water is also tested regularly, but pools almost never need to close.
In June of 1995, I got second degree burns on my hand. While it was healing, my doctor still allowed me to swim, but told me to do it first thing in the morning, when the chlorine had had all night to work. After swimming, I would shower and then apply antibiotic cream to my hand. And, no, I didn't get any infection!