Most of their “evidence” is not positive evidence, but rather what they see as problems with the evidence of the opposing view.
If something casts doubt on the old-earth position, it is seen automatically as evidence for young earth.
For example, a YEC might point to alleged problems with radiocarbon dating methods as “evidence” for a young earth. Or “lack of transitional fossils.” Even if these claims were true (they’re not—radiocarbon dating is extremely accurate and cross-verified and we have a huge number of transitional fossils), they have nothing to do with the verity of the YEC position, but creationists treat them as if they do.
Incidentally, this is the exact same logic our Holocaust-denying friend used in the thread last night. I think it’s interesting that conspiracy theorists tend to make the same logical fallacies as religious fundamentalists.