Is it normal to have your period for two weeks straight while on birth control?

No, it is not. If you have had a couple of times when you missed a pill and then did the "catch up" thing the next day, it's possible you have unbalanced your hormones, and it is also possible that you became pregnant and are miscarrying. It is also possible that your hormone dosage just needs adjusting, there are many variables, but you should see your doctor, as soon as possible, to rule out the scary stuff and fix it if it just a minor tweaking of your dosage that is necessary.