I used to be afraid of my own blood and gore as well, though other people's never bothered me at all. After I got married, had my daughter, had several miscarriages, and then had an emergency c-section at 24 weeks which I watched in the reflection of the overhead lights, I no longer fear it. I wouldn't recommend that as a way to get over it though!
When I want to change a pattern of thinking, what I do is first try to catch myself when I have those thoughts. Then, I change the thought from, for this example, "Oh that is so disgusting! I'm going to pass out!" to "Okay, initially that was a little gross, but really it isn't so bad. I'll just take a few deep breaths. I can handle this." The key is to challenge the old thoughts when you have them and replace them with something new, whatever you want the new thought to be. When you find yourself getting nervous, acknowledge it. Tell yourself that it's okay and you can get through it. You don't need to think of it as needing to change the whole thing right now, but focus first on noticing when you have those thoughts. Then you can work on challenging those thoughts. Just one step at a time.
I'm sure others will have some better advice, but this has worked for me when trying to change my thought patterns. Good luck! You CAN do this!