If you get bit by a person you're garanteed to get an instant and sometimes fatal infection. That's real world. If someone on the street walks up and takes a chunk out of your arm with their teeth...you better go to the hospital and get a shot.
Zombies carry a very active form of the virus(or whatever...the TV show describes it like a fast acting meningitis so it might be a form of bacteria and not a virus) plus they're rotting corpses so it's very likely that getting bit simply results in a very bad infection that is immune to antibiotics. How fast it kills depends on the person so...likely it depends on how healthy that person is at the time they're bit. Cutting off the limb if it's bitten before the infection spreads will save that person which is the same with heavily infected wounds in general. Lee's arm isn't cut off until after he passes out which means the infection has already spread beyond the arm. They didn't do it fast enough. Of course if the infection kills you, you turn because that's how it works in this universe. You turn no matter how you die.