Hello, I'm so sorry you're going through this. I self harmed from the age of 12 and have bipolar. I didn't understand why I cut at the time and found it so shameful that I absolutely refused to discuss it with anyone.
With hindsight, I can see that it was an unhealthy coping mechanism. I used it to release tension and stress, caused by day to day issues like bullying, boys, exam pressures and parental arguments and massively intensified by my illness. My mum was so terrified by my self harm that she used to do unhelpful things like refuse me to lock the bathroom door, make me stand in front of her in my pants to prove I hadn't cut. I used to spend so much effort resisting the urge to cut, so as not to let her and other adults down, that when I gave in I did it much worse because I felt so bad about myself. My cuts got bad enough to need stitches. I eventually used it as a way to punish myself, because I felt everyone hated me for being ill.
I am now a happy adult with a lot of scars. Bipolar is not that hard to manage, but just means that everyday stress will hit your dd harder than most. Respect that, and let her know you're on her side. I felt like a total freak for being sad and dissociated all the time and this intensified my alienation and cutting.
Even with the cutting, she is a teen and will value respect and privacy. Paradoxically, it might be safest to let her have her own space, than try to supervise her so much that she has no opportunity to cut.
Also, I had very few healthy coping mechanisms at that age. Took me many years to learn to stop, slow down, have a bubble bath, even something basic like saying nice things to myself instead of being stuck in a nasty depressed mental loop made a huge difference.
Matt Haig's book "reasons to stay alive" is really great and helped me.
I'm glad she has support. In time she may feel it's easier to be more open with you. My own mum was a paeds counsellor and I didn't talk to her til I was in my mid 20s! Sometimes it's just too hard to tell your parents/loved ones the worst parts of depression. At least in the moment.
My own cutting spontaneously ceased when I got to a certain age, I think because I learnt to make myself a nice dinner, go for a walk or a run, do happy little things to cheer me up so the tension didn't built up like it used to. Now the idea of cutting makes me feel sick. I feel your dd will be okay.