Fighting for a relationship I know is bad for me?

Amy
I'm feeling super stressed, confused and upset at the minute. My girlfriend of 2 and a half years and I split up around Christmas 2016. We lived together and got engaged in October 2016. But things turned sour very quickly when she got a new job that controlled her life. Since we split up, we still spoke everyday and went on dates and slept together. We just couldn't decide what we wanted and kept arguing. But then I found out that she slept with someone she works with in January 2017. I was devastated. We weren't technically together but it still hurt. She begged me to give her a second chance but I was disgusted and cut her out of my life. Since then, I've realized that I am too hung up on her still and decided I wanted to try and fix the mess we had created. I kept getting mixed signals from her, one minute we were talking and agreeing to meet up, then the next she was saying nothing will change and we will never work because everything will go back to how it was Pre-breakup. That made me want to fight harder. She kept asking for space but I couldn't stop messaging her. We met up and went for a drink and kissed. But then I found out that she is "dating" a different girl she works with. She was talking to me and met up with me behind this girls back so now she's hurt both of us. She's told me that she is very confused and thought that it was over between us so she tried to move on by seeing this girl, pretty much as a rebound. But when I changed my mind, she was already seeing this girl so she didn't know what to do. She didn't want to hurt either of us but has. I'm so confused because I just want her back. I know I shouldn't after everything she has done, but I just want our old life back. I'm so lost. She has asked us both for space so she can figure out what she wants, but I don't know how I'll cope if she chooses this girl. Is it normal to want to fight for a relationship you know probably isn't good for you?