If being a supportive ear is something you’re willing to do, but you’re struggling with the constancy of it, one possible way to frame your need for space might be to open up a little about the uncertainty you feel about how to support her, and how you feel bad for not knowing how to help more. I imagine that often, your delay in replying to texts is because you’re trying to respond but don’t know how. Your sister is at the age where people grapple with the fact that adults are often just as clueless and overwhelmed with the world as teenagers, which is made harder by mental health stuff. Maybe showing a bit of vulnerability here may help her to understand that you do care, and that you’re trying really hard to support her, but you have a lot to learn, and that figuring that out will be a process that involves both of you (i.e. I imagine that your sister is not especially good at understanding and communicating what emotional support she needs — she’s only 13 after all— and learning will take time). I suspect that the unreasonable demands that your sister makes (“when they don’t give her exactly what she wants 24/7”), and her responses afterwards may stem from her feeling unfulfilled when she does get what she wants, because a mentally unwell teenager isn’t necessarily going to want healthy things. Hopefully she’ll be able to work through some of this in therapy.

Sometimes if I am struggling with mental health and I message someone, I end up stewing in anxiety as I refresh my phone, hoping for them to reply. That can be pretty harmful, and lead to unpleasant dynamics in relationships. Maybe it would help to carve out time to reply properly. For example, when she messages, replying with a specific time at which you will more substantially reply. This would need to be discussed with her first, and I’d frame it like the space that you are needing is in order to better support her, because “it’s not fair on you if I’m too distracted to properly process what you’re telling me. I want to be fully present, and listening to you, but that’s not always something I can do on short notice”. The challenge that’ll be hard to communicate to her is that, whilst your love for her is infinite, your energy and ability to help is not. I’m really getting a sense of burnout from your post, and it’s tragic that your sister’s insecurity causes her to misinterpret your wish for space — she doesn’t realise that you having that space is a necessary part of being able to support her.

It might be useful to discuss a sort of “tier” system. For example, when a friend who was emotionally supporting me had to take a step back due to burnout, she emphasised that I had blanket permission to call her if I felt like I was at risk (of self harm and/or suicide). Ofc, it’s hard for someone who is in a mental health crisis to gauge how severe that crisis is, but I would hope that a mental health professional could help her to draw up a crisis plan (which would ideally involve self soothing strategies also).

My last suggestion is that maybe it would be nice and useful to carve out some regular time for you two to do something fun together. I remember when I was a mentally ill teenager, it felt like my entire life and all my relationships were consumed by the depression, and the few instances I got to do something nice and normal stick out as bright spots in my memory. Communicating this would probably link into the vulnerability I mentioned above, and being open about how you deeply love her, but you have been struggling because you feel like you’re not doing very well at supporting her. A lot of mental health treatment focuses on reducing the bad stuff (whether that be anxiety, self-injurious behaviours, compulsions etc.), and it can feel like the bad stuff is all there is. Only suggest this if spending time with your sister in a “fun” context is something you genuinely want, and try not to frame this like it’s a pity thing — the purpose of this endeavour would be to benefit both of you, and to strengthen your relationship. It might take a while to find something you’d both enjoy, and you both might have times where you struggle to engage with it, but I think that it’s a useful way to build towards a hypothetical future where your sister is still unwell, but is coping far better, due to engaging with a range of different support.

This comment has thus far largely focused on your sister’s feelings, so I’ll finish by saying that although you feel out of your depth, you’re doing really well. You’re recognising that you’re not coping with this, and that’s good, because your feelings and struggles are just as valid as your sister’s. It can be easy to dismiss one’s own needs when a loved one’s needs appear far greater, but you can only usefully help if you look after yourself too. That’s what you’ve been trying to do, and if it feels like you haven’t done too well at that, it’s because there is no easy guidebook for this. Sometimes when your sister blows up at you, it will be nothing personal, and it’s as if it’s the ill mental health speaking. Sometimes, when she blows up, it’ll be because you’ve legit messed up and it can be hard to discern whether you owe her an apology if her reaction looks identical to when she’s blowing up “irrationally”. Sometimes she’ll say awful, hurtful things that it’s strategically better to shrug off due to her mental health, and sometimes it’ll be better to hold her accountable, because ill mental health may be a reason for acting a certain way, but it’s not an excuse. It’s messy, and it’s tough, so it’s especially important that you are also supported in looking after your own wellbeing.