I'd read that the shroud had probably gone through several monasteries in Turkey and other places in the Middle East before ending up in Turin. It could possibly have gone with Mary, after the Resurrection, then taken up by Jesus's apostles, and their apostles as the Faith spread out from Judea. This could explain why iconographers from various places all painted Jesus in a similar fashion, though they likely never saw each other's works. They were working from the same model; the image on the Shroud.