"that" + "would" = "that'd"?

Is "that'd" an appropriate contraction of "that" and "would"? I say it, but I'm not sure if it's a legitimate contraction in written form.

There are many incidences of that’d meaning “that would” in the Corpus of Contemporary American English:

SPOKEN    208  (2.39/million words)
FICTION   384  (4.7/million words)
MAGAZINE   58  (0.67/million words)
NEWSPAPER  48  (0.57/million words)
ACADEMIC    3  (0.04/million words)
TOTAL     701  (701/million words)

It is most common in spoken English and fiction, so the idea that it’s more for informal registers has merit.

It is certainly acceptable in the sense that any native speaker would understand it. So, I think that would characterize it as legitimate. In formal writing, most contractions are avoided anyway — though if I were somewhere between formal and informal, I would definitely get rid of that'd before I would get rid of it's. So it is on the "more informal" end of the contraction formality spectrum.