Is it normal to pee a lot and feel a slight urge to go pee after you've had a bladder or kidney infection and there's no more infection in you?

Possible: Did you complete antibiotic course? Sometimes it takes time to get back to normal. Most of the time once you complete antibiotic, your symptoms should go away. If you continue drinking lot of water then you may feel reflex to urinate. Needs to consider diabetes as well if your symptoms persist after being treated for UTI long time ago.

Yes, BUT!!: Post void mild urethral irritation, suprpubic discomfort or increase urge & frequency are common during or immediately after treatment. However persistent of symptoms for long time especially after an antibiotic is an indication of partial or unsuccessful treatment & may need follow up evaluation with UA, urine culture & even US for detection of vesico-ureteral reflux or anatomical defect.

Yes: The lining of the bladder can remain inflammed for several days after your infection has cleared. drink cranberry juice and plently of fluids while you are healing. if you don't get better, or get worse, go back to your doctor to be checked out.