According to this article[1] (which has a nice wide-angle view of t-pillars in context), it was radiocarbon dating[1]. You find bits of stuff at the site and the isotopic composition of the carbon-containing material tells you the age based on known rates of carbon-14 decay.
The actual isotopic analysis seems pretty solid.
The problem is that this isn't any ordinary site. The article notes that the site appears to have been deliberately buried. This raises the question of where the samples that were dated actually came from.
This critical review suggests major problems with the older-than-everything-else hypothesis for the site.[2] It notes that at least some of the samples were dated from "fill," or the stuff that was used by someone at some point to bury the site. And that stuff could have itself come from sites much older than Gobeklitepe:
> We already discussed the problem with dating “fills” as opposed to dating “structures”. A fill’s date (no matter how confident we may fill about its actual date) in no way dates structures, as it simply can be coming from soil deposits that are either older or younger than the structure itself. You can fill your home with dirt from your yard, which could be from various geologic strata, some containing fossils from the Pleistocene. This will not make your home a Pleistocene Epoch home. Or you can currently fill a 4th century BC Temple with soil from riverbanks containing live exoskeletons; this will not render the Temple a 2000 AD structure.
Even if the site wasn't deliberately buried, everything hinges on where the fill came from. The base assumption of radiocarbon dating is that no foreign material was brought in. The shakier that assumption, the shakier the claim to the ages being quoted.
[1] https://www.asahi.com/ajw/articles/14487942
[2] https://www.researchgate.net/publication/317433791_Dating_Go...