I cannot really say whether an abusive relationship can become healthy, quite possibly if that person is dealing with certain issues or even hormonal imbalances which make them act out, but I would say in that case they would be very aware of what they are doing and would feel bad about it and would actively want to change. These though would likely be cases of pretty minor abuse, such as high irritability, being argumentative, minor manipulation or control tactics, maybe minor physical stuff, like slapping (not with full force) or throwing non dangerous objects in moments of high frustration. The most important thing is whether the partner recognises there's an issue and wants to change, you cannot force anyone to change their behaviour, you can only change how you respond to it.
I could not even guess at any statistics, but I would imagine that while some abusive relationships may become healthy, it is probably highly unlikely and is essentially the exception to the rule. The general rule would probably be that no, abusive relationships do not tend to turn into healthy ones. People who treat you badly generally just don't care about you that much. They may care about having you around and feel a sense of ownership over you, but that is not the same as caring about
you. As Jicky pointed out, if the person is a narcissist or has anti social personality disorder, and it is causing them to be abusive (it's common for abusive people to have personality disorders), then no they will likely never change. They could change their behaviour if they believe it is not working at getting what they want, which is why these types of abusers are so difficult to get away from. They know when to be amazingly sweet, which completely contradicts the nastiness. Inconsistent relationships are some of the most damaging, and make it far harder to demonise a person and therefore leave. The abuser knows this, and will use these tactics to keep their partner sweet. It's really hard to leave a relationship that has so much control over you, especially when there are feelings involved, but it sounds like getting away would be the best choice for you. There are so many other people in this world, ones who you would get on with great and would treat you how you deserve to be treated. Life is short, don't waste yours trying to make something work with someone who chooses to harm you.
I'm trying therapy on my own but the first session was kinda just her venting about her own single life so I'm not putting much hope into it.
Whoever your therapist is, move on. Her talking about her own life is extremely unprofessional and is a waste of your time. It's pretty much a straight up rule for therapists not to mention their personal experiences. Therapy is not supposed to be a girl chat where you each vent, you are going there for professional support. This should not be the norm, so hopefully you will be able to find a new and better therapist. You may not want to do this, but personally I would put in a complaint about her, that sort of thing is IMO serious misconduct and I would hate to see anyone else reaching out for help and then being put off by someone so unprofessional.
Please stay safe, abusive relationships are dangerous to be in or leave, but there will be support available for you which is there to protect women in this situation. I hope you manage to reach out and find a way to gain control back for your life.