Certainly not normal. It's almost as if the turn signal is taking up all the available electricity. Have the alternator and battery checked.

Did you replace the spark plug wires and plugs? Does this have points and a condensor? If so, check to see if the points burn fast. It could be a dead section in the starter motor. As a spokesman for the obvious, it appears you have a short somewhere. It could even be the battery cables. Maybe the ground...