Is there something wrong with my bladder?

There are muscles called spinsters that open and close to control urine (and other) flow. They seem to be to weak to stay closed in your case. This usually happens at old age but there are other conditions when that can happen. I don't know your medical history, but you definitely need that checked, as it could result in an eventual infection that could travel up your ureters into your kidneys, not to mention a lot of embarassment and adult briefs. You can help strengthen those spinsters by focusing on contracting and relaxing the muscles you use to urinate. Every time you urinate, try to control the flow (stop, go, stop, go...).