It goes into the sewers and to the waste treatment plant the same as anything else. How else could be managed. Contaminated sheets, etc. go to the incinerator at the hospital.

Chlorine (bleach) waste management is a huge problem, being addressed. I think the forest product industry (paper, etc) is the biggest user of fresh water (trillions of gallons/yr) and the biggest dumper of chlorine in waste water. Other pollutants too, from both that industry and laundries (detergent ingredients) are dumped into rivers as well as city sewers going to processing plants. Closed loop systems are being built that extract the pollutants before waste water is released, as well as alternatives to bleach.

A large hospital near me closed their laundry building a few years ago, and contract the work out. I imagine more of that will be happening as specialized equipment is designed to handle huge volumes of contaminated water.