Is it possible to become a nurse?

I have 4 years experience of being a NHS Healthcare support worker, but always had an ambition to become an adult trained nurse. I have worked in both Mental Health and adult nursing and really enjoy both. What makes me doubt myself though, is the fact I can get a bit panicky and pass out when I have had blood taken or the sight of a lot of blood . This hasnt happened at work more so when I was younger. I do think I have somewhat gotten over it, however fear it's always there. I pushed myself and successfully complete the phlebotomy course amd I do this whilst at work when required and I do it successfully. Question is, do you think this is something I can get over to become a qualified adult nurse?