Depo-Provera - can depo cause vaginal bleeding and spotting that lasts for weeks?

Any spotting that you've had in the beginning, normally stops after you've had 2 or 3 shots.
If this spotting is only starting since your 3rd shot, then the way to stop it is to have your next few shots earlier than normal. This is how the doctors in the UK treat it, & it has worked every time for me.
For example, you'd get your next one at 9 weeks, rather than 12/13 (whichever you normally get it at), & then as long as you have no more bleeding, you'd get the one after that, 10 weeks later. If you did still have bleeding, then you'd get it at 9 weeks.
You'd continue increasing it by a week each shot, until you find an interval that means you don't get any bleeding in between shots. For me, I had to have my shot at 10-11 week intervals every time. If I took it later, then I'd bleed before my next shot was due, especially after sex.

The other thing to be sure of, is that you are up to date with your cervical smear/PAP test, which in the UK we have every 3 years (slightly different when we get older). That will rule out some other causes of the bleeding.

If you have any other symptoms, such as abnormal discharge or pain, then you should see your doctor.