Acclimation to BC takes three to six months. It's not uncommon for you to have breakthrough bleeding. I've bled more than not since I started my BC two months ago and I've not had a day off it. My gyn changed the dosage so I'm hopeful mine will sort itself out.
Regardless, talk to your doc.
Breakthrough bleeding and really long periods are common side effects on depo provera. It's actually one of the reasons most women stop using it. Unfortunately, because it's an injectible birth control method, there isn't much you or your doctor can do if you are having unpleasant side effects. You just have to wait for the 3 months to be over and your body to either adjust to it, or to see if it just won't work for you. I used it for 7 years and after the first shot I didn't have any bleeding for most of the 7 years. I think around half of all women taking it have their periods completely stop after 6 months on depo. But it is so hard to predict who will do well on depo. Good luck!