I think you need to use your next therapy session really well (the one where you will be alone, I mean), and try to get everything out, make sure your therapist knows the situation, don't try to hide anything or under-exaggerate anything. Tell her you need to see her regularly, and by yourself. Tell her you want personal therapy sessions and ask her what she's going to do to make that happen.

My best advice is tell your therapist about your bulimia. That really is something that you need to get help for NOW because it honestly can become a lifelong illness and as you already know a life with an eating disorder is no life (I'm recovering now from 9 years of anorexia, trust me that it's worth getting help for NOW, if I could go back to being 16 and get help my gosh I would). Don't let anyone, even your mum, stop you from getting help for that.

Your mum is wrong about it being your fault, none of your mental health problems are your fault. Your mum should definitely understand more, because she herself suffers with depression, I'm really shocked that she is not more helpful to you. Keep fighting for yourself though, it's the best thing you can do right now.

x