"I can't stand the sight of blood" is more common. It's the equivalent to "I can't bear to see blood".
"I can't bear to see blood" and "I'm frightened of seeing blood" can be interchangeable, but there is a different nuance in meaning between the two. "I'm frightened ..." is more about fear. "I can't bear ..." can be about fear, but it could be just an intolerance or distaste about seeing blood. It may not be fear why you don't want to see blood, as the case when you use 'frightened'.