Indeed, I saw a discussion on a Christian website some time ago, in which a doctor who specialises in head injuries explained how dangerous (as well as painful) it would be, there are so many blood vessels at the surface of the scalp. Being a fairly arid country, there'd have been no shortage of thorny plants in Judaea, but if the crown were made of the kind of vicious thorns shown in classic paintings of the Crucifixion, it would have been something of a miracle that Jesus survived to be crucified!
However, the word used for Christ's crown, akanthinon, has a pretty general sense of 'something thorny or prickly', so it's possible that, for example, the crown was actually made of leaves of acanthus, which are prickly, like giant holly leaves, and would certainly be painful, but less likely to cause life-threatening bleeding.