Sounds like your girlfriend is lucky to have such a good guy, and is being somewhat unreasonable. You should not act like a father figure, you are merely her boyfriend. You're doing a whole lot by helping out so much, and shouldn't be expected to pal around with the child too. There's a limit to how much of this an adult can be asked to do; when a kid is only 4, it's babysitting, not a mutual fun thing like (for example) taking in a ballgame with a 16-year-old son of a girlfriend.
Gosh, I'm going to agree with Melissa. I honestly think your girlfriend's expectations are too much! And ANYONE who knows a 4 year old knows they are fickle and funky at times and sure, she has bonded with you but when mom is there she wants to be with mom. My kids were the same way. My kids grew up with their father in the house-- my youngest preferred me for YEARS. Not because he wasn't bonded with his dad but he was just a Mama's boy.
Anyway, I think your girlfriend is very lucky to have found someone who will care for her child the way you do. And she is ruining it by putting even MORE pressure on it. This will get old. I hope you stick it out but you do not need to do anything else other than be genuine with the child and show her love when you feel love. Your girlfriend is not being reasonable but probably also won't respond well to being told this. So, just tell her that you care very much for, love her child, etc. and that you are doing the best you can do.
good luck
Sounds like you do enough already with your girlfriend's daughter. Just sit down with your girlfriend and have a heart to heart talk with her. Communication is the key. Tell her how you feel which may get her to tell you how she feels. Good luck