Your sense of smell is 10,000 times more sensitive than your sense of taste. That’s why you can detect chlorine in water even when it’s only at 4 parts per million—a level far below the safety limit. You’re not being dramatic; your nose is just a very thorough health inspector.
Also, if you ever travel to Paris or Barcelona, don’t be shocked by the strong chlorine taste. Many European cities use higher levels of it due to older pipe systems. Pack a portable filter. Your taste buds will thank you.