There's definitely just some variance in terms of how much it bleeds and whether it hurts. I haven't really noticed any pattern in terms of injection site; I think there are a lot of factors. Holding the vial and maybe rolling it between your hands can help warm up the liquid, which makes it a little bit easier to both draw and inject.
The needle size will affect these things too -- the thinner the injection needle the less blood I've seen, but then it takes a little longer to inject. Do you know what gauge you're using to inject with?
I found the whole process very stressful for the first several weeks, but eventually it just became routine!
Witch Hat Atelier has girls learning magic by being taught it. It's not really the magical girl genre -- and their mentor is a guy and there are plenty of male witches in the background -- but the overall focus is on a small group of young girls learning magic.
Not sure if that's really what you're looking for, but its at least adjacent! (Also, the art is lovely.)
e: Oh, and there's an upcoming anime of it, here's the trailer