I had a partial paralysis of my face after the dentist injected me and it lasted for 3 hours. Is this normal and what causes it?

Bells Palsey. Temporary facial paralysis can happen after local anesthetic is used. It is not normally the intent of the dentist but if the facial nerve is anesthetized your face will droop for a short time. Usually the dentist is trying to anesthetize the sensory nerves but sometimes the motor nerve is anesthetized causing the muscles to relax.
Transient paralysis. after local anesthetic are common. Such facial paralysis is transient and temporary in nature and is of no lasting consequence.
See below. It is normal for the face to feel numb and to the pt enlarged to the extent that it feels worthless and the time frame is not unusual. Paralysis sensation is likely due to part of the facial nerve beeing anesthetized as well which can happen. No real need for concern especially because it only lasted about 3 hours.