Believers don't go to any lengths to prove them authentic. They can't be proven authentic. Furthermore, relics are not seen as necessary proof for historical events. That's a completely incorrect understanding of relics. They are principally objects of devotion.
They are handed down and popular devotion leads us to believe that these are things associated with the life of Christ or his saints. For that reason we cherish them, just as you might cherish a keepsake of a loved parent or relation. However, just as your grandfather's Bible is not necessary to prove that he really existed, so the burial cloth of Christ is not essential to believing in his resurrection. Nevertheless, on the belief that it is, we venerate it because this is something which covered the body of our Savior. Out of love and respect we guard it, just as you might guard a memento of a loved one.
The disordered "yikes" mentality which surfaces when relics of Jesus' time on earth are discussed, is truly comical.
What happens to those that idolize these items when they are shown to be fakes?
Do they question their faith? Do they blame their church for misleading them? Do they finally realize their devotion shouldn't be of some object?