Is it normal to experience significant pain while breastfeeding?

No. Mild discomfort is not unusual but severe pain can be caused by mastitis, cracked nipples or difficulties with latching on.
No. You may experience pain, sometimes lika a pinching sensation as your baby latches on and pulls at your nipple, it may be painful for about a minute but it should stop, if it does not, may be try repositioning the baby. If it still persists there may be something else going on and it would be prudent to consult your doctor.