The result of the Carbon 14 dating test carried out in 1988 gave the date of origin as 1260-1390. But unfortunately, the decision about which bit to test was only made on the spot when they actually came to cutting a bit off the corner. The bit which was tested, and which produced the dates given, came from the corner which has been repeatedly handled by hundreds of people whenever it has been put on display. It is believed to have been contaminated with loads of bacteria, which would have contributed extra carbon and thereby made the C14 test result unreliable.
There were numerous other avenues of investigation which the programme discussed, including blood, sweat, photography, etc.
There is a distinctive pattern of four holes on one part of the shroud - the same pattern also appears in an illustartion in an 11th Century Hungarian illustration, and is likely to be a picture of the same shroud.
The provenance of the Turin shroud is not very well documented, but there is another cloth which has got a well-documented history going back to the 6th century. It is believed to have been wrapped around a dead man's head as part of some kind of religious ritual. The interesting point about that is that the Turin shroud and the other cloth have both got blood type AB on them - i.e. only 3 per cent of the population - and may have been used on the same body.
The main conclusion of the programme was to be inconclusive, and to suggest thast new C14 tests need to be done on other parts of the shroud - and other tests e.g. to test for the presence of silver sulphide which may have been used in the formation of a photograph (one theory is that the Shroud is a photographic negative made in the 15th century using silver sulphide to fix the image on the cloth).
I think that the programme said there is more info about it on www.channel4.com/belief (although I haven't checked it out yet).