K.D.
According to Babycenter.com, the average 2-year-old needs 13 hrs of sleep, 1 nap about 2 hrs and 11 hrs at night, so it sounds like she's not getting nearly enough sleep. I suggest getting her on a schedule (if she's not already) and stick to it. My 2-year old goes to bed at 8-8:30 and sleeps until 6-7 am and takes a 2 hr nap from 1-3 pm every day. She went through a stage where she was waking up really early (5 am) and waking up at night a few times and it lasted about 2 months, but now she's back on schedule and the yelling/screaming/tantrums/etc have greatly decreased. I also suggest that you stop bringing her to bed with you as it will probably just become a habit that's hard to break later (and you say that she's not sleeping when you bring her to your bed anyway). When my daughter started waking up at night crying, I would just go to her room and tell her it was still time to sleep and she'd go back to sleep. I don't think you're doing anything wrong, but it's just a phase a lot of kids go through and that you have to tough it out. You could also try a night light if she seems scared or a sippy cup of water in her bed if she's waking up and is thirsty.
Good luck and I hope it gets better real soon!