There is growing suspicion the swatch used in the carbon 14 analysis may have been taken from a part of the Shroud that had been repaired by nuns after a fire. There is too much evidence for authenticity to simply accept the theory that the Shroud is from the 13th century. For example, the weave of the cloth is first century. Pollen on the cloth derives from the environs of Jerusalem in the first century. The patterns of blood match the sudarium of Ovieto--which had been venerated long before the 13th century. The coins that show up covering the eyes of the shroud date back to the first century. There are also paintings depicting the Holy Shroud which date back to before the 13th century.
I've not kept up with this with the level of detail I should, but I know that the supposed "disproving" of the Shroud is not as solid as was once thought, and there is still considerable room for scientific reasons to think the Shroud may be authentic. I remain open on this, especially as one way or another it would not affect my faith.