Sounds like she needs some one on one time with her mother alone; your girlfriend. I would encourage "mother and daughter" nights or lunches or shopping days or spa days. Give her the time alone with her mother and see if that works.
I feel for you because you are in a difficult position being you are only the bf and not the child's stepfather and she is probably resenting the fact that you are disciplining her which is more of a role of a stepparent and even then limited.
The child should ONLY be disciplined by her mother, not you. If you have a problem with the child you should talk with your girlfriend about it and not directly address the child. That's the BIG mistake you are making.
Leave the parenting and disciplining to her parents and take a step back.
I find it a bit intrusive for you to be reading the child's personal diary being you aren't her parent. You should have been ONLY stripping the beds not snooping. I can't agree that was ok. Sorry. I don't think you would like it if she went into your personal things. Just give her some mutual respect regarding that.
How long have you been seeing her mother? Sounds like not long. You are putting yourself in a stepfather role and I can't say I will agree with that.
This child has alot to cope with....a father bringing "this girl and that girl" into her life plus her mother is with you and she feels you are snuffing all her mother/daughter time out. Then you are taking on the role of stepfather when you aren't. Very confusing and scary for this child in particular. Have her mother talk with her alone. She probably feels every time she turns around there's you with her mother. Give her that "alone time" with mom.
I am not saying a 13 year old should be "running" the show and telling a parent who he/she can see or what he/she can do, but her mother should be sorting this out with her daughter alone. If you two were married, my opinion would be different.
You are not her dad nor her stepdad, and that means, sorry but no matter how much attitude she cops, you have no right to shout at your girlfriend's child. She's going into puberty and needs her mom. No matter how much you like sleeping with her mom, you are acting like the kids are a barely tolerable necessary evil, as in, if I want the mom, I have to attempt to get along with the kids. You aren't acting like you have genuine care and concern for them as people. (If you were the guy who had sat by her bedside all night when she was three and had a fever, my opinion would be different.) Poor kid, she has a father who is exposing her to girlfriends, you are there taking space that she needs, right at a time when she's getting hit with hormones. She's going to be out trying to find a boyfriend soon to give her the unconditional love and undistracted (and undramatic) attention she needs, if she can't get it from her mom. We hear from so many significant others of people with kids, and the questions are never "How can I understand what this child is going through and help?" but always "This kid is being a pain in the behind, how can I get her to straighten up and fly right?" Hey, she's 13. She's been kicked out of one secure family nest by virtue of her parents splitting up. She has no security of relationships; dad's screwing around with one person after another, and mom's got a boyfriend but not husband. (Sorry.) She's scared and lonely as her body is changing. You are sitting next to her mom when she wants to chat mom-to-child. You can't help it, you're in the way, at a time when she really needs support.
Give her and her mom some time away from you. Sorry.