Was your husband a C-section baby? For some reason, people born via C-Section have shown to be more cavity prone. Theories out there think it may have to do with a bacteria from the birth canal that somehow helps with developing stronger tooth enamel?? I don't know exactly what it is, but my brothers and I were all C-section babies, and despite the amazing dental hygiene we grew up with, all had cavities.
okay, back to your baby. While I totally commend what you are doing and trying to instill in her-- perhaps you need to back off just a little. Regardless of her physical reactions, mentally, it's going to become a power struggle. That could have long-term ramifications about her attitude toward teeth brushing more than not doing it properly.
Have you allowed her to "brush" her own teeth? She's definitely got more teeth that need to come through, so the bleeding could be from that. Allowing her to examine the toothbrush, help put the toothpaste on it, etc., might help her feel less freaked out by the experience.
My oldest daughter is 2.5. She LOVES to brush her teeth now, but it took a while to get her there. The thing that helped the absolute best was completely unintentional: She watched ME brush MY teeth one day. That was all she needed, she saw me doing what she was "fighting" and became much more interested. When my mom (dental hygiene fanatic!!) came to visit after baby sister was born, she and my daughter would brush their teeth together. My mom sat on the stepstool in the bathroom to be at face level with her, and told her to do "exactly what I do". She actually brushes SO much better now after a few nights of that.
Now that my daughter is old enough to understand what "going to the Dentist" is, we sometimes play dentist when brushing our teeth. This is my ploy to sneak in and get a few good nights of brushing each week. I tell her "I'm the dentist, let me see those teeth" and then will say things like, "oh my goodness, look at this sneaky tooth--he needs me to really clean him up!" and then I'll just brush away. I even drape the towel over her neck/shoulders like at the dentist office.
Make it fun, keep it light, get a few board books from the library about teeth/brushing/dentists, MODEL for her, let her feel more "in control". If you're genuinely concerned about the gums, take her to a pediatric dentist.
We took our daughter for her first "dentist visit" when she had just turned two, and I expressed concerns about not being good at "really brushing"... the hygienist said that really, brushing counts the most once they get those 2 year molars in, those are the more cavity prone ones. Of course she also said to avoid candy and juice!