“Is it normal for an implant to hurt?”

You should consult with the doctor who placed your implant.

Discomfort is common after dental implant surgery and should be expected in the immediate postoperative period. Discomfort should get better day by day and it is not uncommon to have some discomfort that continues to resolve 1 week after the procedure. Your recovery will also depend on whether or not any other procedures were performed at the same time - i.e. tooth extraction, bone grafting, bone graft harvest, sinus augmentation.
If you are having pain that is not relieved with pain medicine, I would contact the surgeon who placed your implant or present to an urgent dental care clinic in the event they are unavailable. Furthermore, if you have discharge or signs of infection, you will want to be evaluated on an urgent basis.
In the meantime, optimize over the counter medicines (taking into account other medicines you may take, your medical history, and allergies) and any other pain medicines that were prescribed.

Wait 10 days