I think what did it for me was the fact that they said the only other fabric they have ever found that was woven the same way came from a settlement dating to the first century. Once it dates back to that time frame, all the wounds on the body correspond to those Jesus suffered. Also, if someone was going to preserve a burial shroud from that time, it only makes sense to me that it would be Jesus’s. I don’t know who else would rate high enough for a person to unwrap a shroud from a body and keep it.
They also said that there was a cloth in Spain (I think) that is purported to be the head wrapping and a sister cloth to the shroud. That fabric has been documented back to the 6th Century and has the same blood type on it as what was found in the shroud. The blood stains on this cloth also match up with those on the shroud. As for the carbon-dating, they believe that the corner cut for the dating was one of the areas that was held onto when it was being displayed in public in the middle ages.
There was one person on the show who thought it was a forgery. His theory was that it was the first photograph ever made. But to make his theory work, the cloth had to be soaked in a silver solution and then a body was strung up and reflected through a lens for three days to get the image to appear on the cloth. That seems awfully complex for someone in the middle ages. I would think that if someone wanted to forge it at that time, they would have just painted it.