What's Actually in Tap Water?
Tap water is treated to be safe to drink, but it commonly contains chlorine (used to disinfect it), minerals that make water "hard," and trace amounts of other substances that vary by region. Chlorine is the one most people notice in the shower — it's why some tap water has a faint pool-like smell.
For drinking, treated tap water meets safety standards in most developed areas. The reason people filter it anyway usually comes down to taste, smell, and skin comfort rather than acute safety. In the shower, chlorine and hard-water minerals are the usual suspects behind dry, tight-feeling skin. If you're curious exactly what's in yours, local water utilities publish annual water-quality reports for their area. More in what a shower filter removes.
Try a BHG shower filter to reduce chlorine in your shower.