When we go swimming and spend extended time in water, do the little bugs who live on our skin suffocate and die?

I am not a microbiologist. However I do know that bacteria are pretty resilient, and most pathogens and parasites can live for at least a few hours outside of a host. Plus your skin is really not all that smooth at a microscopic level, there are all sorts of little nooks and crannies to hold onto.

Drowning wouldn’t really be a concern for them, imo, but additives like chlorine are specifically meant to be disinfectants.