I puntured my palm with a nail center mass now extreme pain when bending center 2 fingers is this normal?

No, not normal.: Pain is your body's way of notifying you when something is wrong. "Usually" the worse the pain, the more serious the problem. If you punctured your palm with a nail you may have injured an underlying nerve, muscle, tendon, or bone. This can cause extreme pain, and there is also a risk for infection. I would be sure to see a Dr as soon as possible and make sure your tetanus vaccine is up to date!