Please advise if it's normal to not be able to feel almost half of your face after being injected by anesthesia (for tooth extraction)?

Local anesthesia: Injections of local anesthesia for dental treatment not only numb the teeth but surrounding tissues. Especially using what we term a "block" injection will numb large areas of the face. It usually wears off in about 3-4 hours but may last longer depending upon the specific local anesthesia used.

Numbness: Yes it is especially if it is a lower tooth to be extracted. The numbness generally wears off after several hours. If the numbness persists then you need to return to the dental office asap to determine the cause.

Anaesthetic Effect: Block injection can cover more areas of innervation, therefore, is more effective in preventing pain and discomfort during a dental procedure - if this is what you had. I hope you heal soon!