One thing you might do is try to separate the "reasonable" from the "unreasonable" fears. Then try to respect the reasonable ones (change the circumstances) and investigate and adjust the unreasonable ones (change your perspective).
Reasonable fears (say you find yourself doing 110mph through traffic) are telling you to change the situation... slow down. That's just healthy.
Unreasonable fears (say you find yourself reacting uncontrolably to somewhat risky but generally "safe" situations) are a whole different ball game. These are the subject of tons of studies and theories. I (personally) think they are just telling us there is a difference between what our conscious mind knows, and our subconscious mind believes.
For instance, I don't like heights... period. I can walk comfortably on an eight inch girder, a foot off the ground. Hell, I might even dance on it. Move that girder several stories up, though, and I don't care for it at all.
Same with deep dark water like the ocean. I could let myself get to a panic situation there because my subconscious thinks there's things down there that will get me (Jaws). But, if I can get in, and get down near the bottom with some visability, I chill out and actually can get real comfortable. For me, visibility seems to be a key factor. Also, though, is the unreasonable fear of getting eaten. That, I deal with by just imagining a big shark eating me in a couple of bites and asking myself, WTF.... dead is dead and everybody dies, right? Then I quit making a big deal about it.
With your situation with bikes, I'd try to discover specifically what causes the reaction. Is it traffic... speed... catastrophic mechanical failure (front tire blow out)... death? To deal with fear, you usually need to name it and face it.