Emotional intensity? Yeah, we'll give it that. There are even some people getting killed, which tends to happen when you're portraying an extended gang war. But West Side Story was made while the Hays Code—formal censorship guidelines adopted by the movie industry—was still in effect, which meant you couldn't get too graphic. That meant the violence stays pretty chaste; no bloody brains on the wall as the bullet-ridden victim slides slowly to the floor. You get shot or stabbed and you fall cleanly to the ground.
As for sex? As the West Side locals would say, fuggetaboutit. Any sexy stuff is just suggested and happens off-screen.