Do you have a shower water filter?
You probably make sure that your drinking water is filtered because you know that you don’t want to drink chlorinated water. Some people even shower after swimming to rinse off the chlorine from the pool (which is pointless because the chlorine has already been absorbed by the skin and there is probably more chlorine in the shower water, unless it is filtered).
Chlorine is absorbed very quickly and easily by the skin, especially when it is heated. Heat opens the pores and relaxes the skin. The heat in a shower makes it easier for the chlorine to escape the water and become a free gas that we breathe in as well.
Chlorine can cause all sorts of problems, such as dry skin and hair. It can irritate the eyes and the ears. Breathed in, it can irritate the breathing passages, especially if you already have breathing challenges.
If you would like a recommendation, the shower head I use does a great job at neutralizing the chlorine in the water so that your shower water is clean and safe.
Actually getting clean in the shower