It is normal for my leg to hurt at the site of my insulin injection?

Check for infection. If you are having pain at an injection site, you should call your doctor. Your doctor will want to know if you have any signs of infection such as fever, redness, swelling or hot to touch at injection site.
No . Insulin is given subcutaneously. Your whole leg should not hurt. Some could have temporary irritation located to the site of injection, particularly with Insulin pump infusion sites, but your situation sounds different. Please see your doctor.