I went through something similar to this at around the same age.
My family moved from England when I was 7 and I went through the culture shock that your sister did as well. The Aussie kids were so much more outspoken and outgoing and I was much more quiet and reserved so there was definitely a culture clash.
In my early teens, I developed a severe case of social anxiety disorder (minus the cutting and self harm that your sister is experiencing). I was eventually admitted to an adolescent mental health ward because I had a nervous breakdown. I have recovered now though (this was 5 years ago).
Maybe your sister’s case doesn’t warrant severe intervention like it did with me, but I know that people are usually admitted because they are a harm to themselves or others. Those facilities check your bags when your admitted and make sure that no objects you’re carrying are sharp or otherwise dangerous and even the furniture has a lack of sharp corners or places you might be able to hang yourself from so the risk of self-harm is greatly minimised (sorry if that was a bit too graphic).
I live in Queensland, so I’m not sure what state you are in, but here if you want to pursue that course of treatment, you should pack a bag with clothes and go to your local public hospital emergency room. You might be waiting several hours and you might have to be transferred via ambulance to a different hospital. Also, when you get there, there might not be enough beds so keep that in mind (Queensland is notorious for lack of adolescent mental health services in the public system).
Otherwise, I suggest you talk to your parents about getting her a referral to a psychologist and psychiatrist (a psychologist conducts talk therapy, and a psychiatrist prescribes medication). That can be done through your GP. I know your sister is currently having talk therapy at school, but when I was in high school and I had sessions with a school counsellor, I always found the advice to be quite general - basically things that my psychologist was covering with me anyway but in a very brief way and the advice that they gave was not always helpful. That’s just what I found though, she might find counselling/school talk therapy sessions more beneficial than I did.
As for talking to your sister, try not to dismiss her feelings by saying things like “you shouldn’t care what people think about you” or “just, relax”. Instead, just try to listen. Don’t try to suggest “fixes” to the problem either - sometimes you just need someone to tell you that what you’re experiencing isn’t that scary. Of course, if she asks for advice you should give it.
You mentioned that she’s having trouble studying. It also might be beneficial if you helped her with her schoolwork, or if you sit down together and help each other. When I was going through anxiety, I found it hard to get the motivation to do schoolwork and I couldn’t focus on it as much. There was nothing worse than sitting in a quiet room trying to do it myself.
On that note are there any support systems at her school that allow her to get help with study (aside from the school counsellor)? Tutors? Special ed staff dedicated to helping with schoolwork? As much as I didn’t want to be seen as a “sped” I found going to the sessions with those teachers/tutors helped me. It may be worth going to a psychiatrist so she can get a diagnosis and have access to more support at school. She doesn’t have to tell any other kids at school that she sees a psychiatrist/psychologist, or that she goes to extra tutoring sessions (if they’re are any), but it will definitely help in the long run.
I hope this helped OP, and I also your sister gets through this rough patch in her life. :)