I'm sorry your DD and you are going through this. I hope our story (me and DD's) will help.
One afternoon, when DD was in Y9, school phoned to say DD had been self-harming by cutting her arms. She had begged for me not to be told as I would be angry (I wasn't) but the pastoral person said I would be concerned (I was, very). Before DD came home, I read up about self harm online and learnt that it's a way of expressing something difficult that can't otherwise be expressed. Therefore, as parents, we need to help them find a different way to express difficult feelings. What we mustn't do is ask what's troubling them ie respect their privacy or ask to see the cut marks.
As soon as DD came home, I cuddled her lots and sat down with her on the stairs. All I said was 'We need to find a different way for you to express whatever it is that troubling you. Will you let me make an appointment for you with the GP? I'll come in with you, then I'll leave so you can talk to the GP by yourself.' She agreed. The first appointment went as planned, she had a few more with the same GP and that was it.
She's 22 now and in her final year at university, doing very well, really sociable, and with a fantastic boyfriend (fellow student). As far as I know, there has been no recurrence. At an appointment of my own with the GP, I said to the GP that I knew he couldn't talk to me about another patient (thank-you Victoria Gillick) but that I wanted to ask him two questions which required yes or no answers. He said I could go ahead. I asked 'Did my daughter talk to you about self harm?' and 'Based on your conversations with her, is it reasonable to work on the basis that she is nolonger self-harming?' The answer was yes in both cases. Eight years later, I do not know what she was troubled about (I would never ask and she hasn't told me). She has faint scars on her arms and happily wears sleeveless things and bikinis.
I was told by school that there is often an element of copy-catting with self harm and that year 9, which I think your DD is, is the year when girls in particular can have a hard time. I learnt from another school, at which I was the safeguarding governor, that cutting in Year 9 was rife that year in our city ie more copy-catting evidence. My DD in Y8 had a sort-of-boyfriend and she suspected him of cutting, something that may or may not have been relevant.
I hope all this gives you are starting point. In summary, be empathetic but not prying, don't make a big song and dance, suggest GP as a starting point, be a facilitator but don't see yourself as the solution.