Do talismans actually have supernatural powers?

Depends on what you believe. A Christian will tell you that the crucifix has supernatural powers while an atheist would tell you they don't. Likewise, someone who believes in and uses talismans (of which, the crucifix would count) would say yes, while non-believers would say no. Its essentially up to you. However, realize that there is NO scientific evidence that anything "supernatural" exists, period.

Additional information:

A very good example of how talismans work can be found in the movie "Dumbo." The little elephant is told, and believes, that his ability to fly comes from a magic feather. When he loses the feather, he plummets to the ground. He has to transfer his belief in the talisman (feather) to belief in his own ability.