The shower scene stuck with a whole generation of moviegoers.
One of the most disturbing things about that scene was the very sexualized way in which it was handled. The fact that Norman just watched Marion undress, the close-up on her stomach, her heavy breathing right before she died... Does This Remind You of Anything?
As well as the distinctly phallic silhouette of the knife, the ejaculatory spurts of blood. It's classic Hitchcock symbolism.
Janet Leigh, after playing the woman in the shower sequence, never took another shower for the rest of her life. (Supposedly.)
It's not watching the shower scene itself that is uncomfortable, it's the seed you have planted your mind the next time you take a shower alone, and how vulnerable you feel. The whole tiny box / nowhere to run thing doesn't help, either. The noise of the shower and the opaque curtain, allowing just anybody to sneak in, if they're careful enough... it's Nightmare and Paranoia Fuel all in one!
The very ending, with Mrs. Bates having completely taken over Norman, and him sitting there in the police station with the blanket. The voice over: "They're probably watching me. Well, let them. Let them see what kind of a person I am. I'm not even going to swat that fly. I hope they are watching... they'll see. They'll see and they'll know, and they'll say, 'Why, she wouldn't even harm a fly...'" Combine that with the Kubrick Stare and other matching facial expressions Norman has in that scene, and you've got some really creepy stuff. Just before fading the scene the dead mother's face is superimposed over Norman's. It's hard to see, but really adds to the creepiness.