Some scientists in Italy claim to have done so by showing how the image might have been produced using 14th-Century technology.

Of course, there was no need to go to all that trouble. There’s a much more authoritative source which debunks the Shroud of Turin.

From the Gospel According to St. John, Chapter 20:

6. Then cometh Simon Peter following him, and went into the sepulchre, and seeth the linen clothes lie,

7. And the napkin, that was about his head, not lying with the linen clothes, but wrapped together in a place by itself. [Emphasis added.]

According to the Biblical account, the cloth used to wrap Christ’s head was completely separate from that used to wrap his body.  Therefore, since the Shroud of Turin purports to be a single piece of cloth used to wrap both head and body, it cannot be the cloth used to wrap Christ’s body.

(Of course, this debunking only works for people who believe in the Bible, but I think the number of people who believe in the Shroud of Turin but not in the Bible is relatively small.)