The practice of offering turkey to God was popular in the time of King David, which is why it appears so often in Psalms. It derived from an older practice of offering turkey to one's parents: people thought, Why offer it to our parents? We can offer it to God!, and did so. The older practice of offering one's parents turkey is alluded to in the verse in Genesis (37:3):
וְיִשְׂרָאֵל אָהַב אֶת יוֹסֵף מִכָּל בָּנָיו כִּי בֶן זְקֻנִים הוּא לוֹ וְעָשָׂה לוֹ כְּתֹנֶת פַּסִּים
En hamikra omer ela "Darsheni!". This verse makes no sense on first read — what does having a beard have to do with a shirt? — and must be read at a deeper level.
"אהב" is clearly an acronym for "אכל הודו בלילות", "ate turkey at night", so the verse reads:
Jacob ate turkey, from all his sons, at night with Joseph. Because his son [=Joseph] was bearded, he made him a shirt of stripes.
The "shirt of stripes" that Jacob made for Joseph was actually a sort of bib to prevent the barbecue sauce from getting in his beard. It had stripes so that stains would not appear as visibly as they would on a bib that was all one color.
Later, after the time of King David, the custom evolved further. People stopped offering turkey wholly to God and started eating parts of it, as with a korban sh'lamim. Eventually, they stopped offering it at all, and just ate it themselves, from which derives the modern practice of eating turkey on Thanksgiving.