I'm a mental health nurse (I don't work in CAMHS though), and I self-harmed for many many years, which was my inspiration for becoming a nurse!
Self-harm has lots and lots of different functions- self punishment, liking the blood, pain to distract from emotional pain, etc. For me personally it was because I truly hated myself and felt I deserved it, and tbh it also became addicting and some days I did it just because it was part of my routine.
Firstly, a bit like with alcohol, you won't stop someone self-harming unless they're ready to. You're better supporting someone to ensure they use sterile blades/have access to good materials for wound care. There are places that are 'safer' to cut, such as tops of thighs.
That's not to say you can't have a really good conversation with her about self-harm and try and get her to agree to a couple of things. For instance, locking her blades in a box in a locked drawer. The physical barrier gives you more time to think and requires more planning, as self-harm is typically quite impulsive. If you're close, encourage her to come and sit with you until the feeling passes, doing whatever she needs, even if it's just sitting in silence. The thing with self harm is, it happens when your emotions are exceptionally heightened, but your emotions never stay that way forever, they always come back down. You could also do things like encourage her to wait 15 minutes first, then if she still feels she needs to self-harm, she can. There are lots of other things you could encourage her to do first, such as draw on herself in red marker, twang an elastic band, hold some ice, or etc. Different things work for different people. There is also an app called Calm Harm- you can kind of select why you feel like you want to self-harm, and it will suggest activities for you. If she relapses, it's important to let her know she's not a failure and help her to not judge herself, as recovery from self harm is not linear. If I'm feeling very sad, my first thought tbh sometimes is still to SH, but I don't anymore. You totally can recover, so please hold that hope for her. She will probably feel lots of shame, so please keep trying to talk to her about it. Don't shy away from it.
Make it very clear that you are not trying to shame her and that you understand it is a way for her to cope with some probably very difficult feelings. Frame it that you love her and that she is loveable and that her body does not deserve hurting or punishment, which is why you'd like to help her.
I'd also encourage her to write about it after it's happened. Get her to think about her feelings and what led up to it, how she was feeling when it happened, and how she feels after. SH often works initially, but then just makes you feel crappy later on and the endorphins wear off. If she doesn't really know why she self-harms, this might help her insight there and to spot what kind of situations are triggering for her.
Sending you lots of love OP. Self-harm is horrible for the person (it's feels shameful, it's not something we're usually proud of) and it's horrible for families to see someone they love harm themselves like that.
I was a chronic self-harmer, and I made it out the other side. Your daughter can too. I don't feel ashamed anymore. It was there when I needed it, but I don't anymore.
Good luck, and please DM me if you need a listening ear. I'd never judge, as I've been there and I've heard it all at work!