Faint at Sight of Blood

I've always known that I don't handle trauma well and that I get faint at the sight of blood or even a vivid recount of a traumatic situation of someone that I know well. I worried about that before becoming a mother but I was told that it's different when it's your own child and that somehow you just handle it. Well I've recently discovered that that is only partially true for me. My daughter is 19 months now and she has busted her lip two times. Both ended up being mild injuries with just a little swelling and a tiny scab but WOW, the bleeding would suggest otherwise! The first time I was home alone and I did OK at first. I took her to the sink and held a towel to her mouth until it stopped bleeding. But once it stopped it hit me. I got dizzy and light headed so I sat on the couch. I really felt as if I was going to pass out. So I got up took some deep breaths and told myself out loud "I can't pass out!" I wasn't sure what was going to happen so I walked over to my next door neighbors house. After sitting with him for a bit outside I felt better. The second time my husband was home. We were all sitting on the couch watching TV and she just takes a nose dive off the couch. I picked her up and handed her to my husband and the bleeding started. He took her into the bathroom, I didn't feel woozy right away but I was hesitant to go in there. I started down the hall and asked how she was and closed the door. This of course made me panicked that it could be really bad. I did OK until after he opened the door (once the bleeding stopped and he had her cleaned up) but there was some blood on his shirt and then the dizzyness started. He asked me to hold and rock her so that he could get a good look at her teeth and I had to sit down. I was OK after a few minutes but this really worries me. I don't know how I will react in a situation that is much worse. I hope that I can at least continue to respond immediately to the situation but I doubt that my reaction is going to change. I feel really powerless in this situation. I don't know how to make myself stronger and able to handle trauma without feeling faint. My baby needs me to not pass out if she's hurt, especially if no one is there with us. I've wondered if there is any type of therapy could help, maybe this is realted to past trauma that I've experienced. Does anyone have any advice or similar experience?