Look up the "Sudarium of Oviedo". This is the facecloth that went with the Shroud. It has been in Spain since 600 AD (700 years before the Shroud came to France). The bloody marks on the facecloth, and their DNA match those on the Shroud perfectly.
You may be thinking of the carbon-dating, which indicated a Mediaeval provenance. However the dating was done on a piece of the Shroud used as a "gripping point" during public viewing. For centuries the Shroud was displayed to the public, and held lengthwise by a dozen prelates at least once a year - and their hands weren't all that clean. Its easy to see how this could throw out the C14 dating. (Also, of course the entire Shroud was suffused in woodsmoke during a fire in 1532, further corrupting C14 data).
There is a book out called THE RAPE OF THE SHROUD OF TURIN by William Meacham, who lives in Hong Kong but travels often to Italy.
It has been shown that the C14 sample site adjacent to the Raes Sample site were both taken from an area that had been patched in the 16th Century by a technique called French Invisible Reweaving. This technique literally connects new thread to old thread and reweaves the cloth in situ.
The sample left over from the 1988 C14 test has been examined closely and has been found to contain approximately 45% new material that has been fullered by a medieval technique, retted in a distinctly different manner than the main body, dyed with madder root to match the aged linen of the main body of the Shroud, spun with a "Z" twist opposite to the "S" twisted threads found on the Shroud and the Linen contains inter-spun fibers of cotton which is not found on threads from the rest of the Shroud. Finally, the threads of the patch contain high levels of Vanillin which is entirely absent from the Linen of the rest of the Shroud.
Since the sample was flawed, the test was flawed.