Is it normal to have back pain in the area 4 weeks after spinal tap?

No. Could be an infection ...Go see the dr that gave you the tap asap.
Maybe not. If the pain began after the tap, and that it persists and worsens, it may be the sign of a problem such as infection. Contact the doctor who did the spinal tap, especially if you are having other symptoms. If you have fever, chills, headache, light sensitivity, seek attention right away.