Hi J.,
I think it's normal too :). It's just one of the personality types out there.
My son is similar in many respects, and when he was your daughter's age, he was terrified of all other children and would just shut down and withdraw in social situations -- I can't tell you many nights of sleep I lost and how many gray hairs I grew over that!
Now that my son is four, though, he is doing much, much better. He actually has friends, including one best friend, which is unbelievable given where he was just six months ago. He used to have this terrible fear of the wind, and that's now gone, thanks to a series of "science experiments" I did with him demonstrating that the wind can only knock over light things like paper, not heavy things like people. He still *drastically* overreacts to having his hair washed, and he's still generally a shy, cautious kid, but these are ASPECTS of his personality. They used to BE his personality.
Anyway, sorry to ramble. What I've learned from experience and self-education is that shy, cautious kids can often have a rough time during the toddler years and early preschool years, but they often outgrow the worst of it and grow up to be sweet, lovely, thoughtful, reflective people.
The only bit of advice I feel confident giving is that it's often much more effective to express things in the positive than in the negative. "Don't be afraid of the lawn mower" doesn't work anywhere near as well as "The lawnmower is nice. It only moves when people push it," if that makes sense.
Good luck,
Mira