The cartoon is great.
I looked up "Turkey" in an etymological dictionary and this is the entry:
http://www.etymonline.com/index.php?term=turkey&allowed_in_frame=0
So, to summarise it: There was a big bird on Madagascar that had a comparable stature, but was a different species. And it was imported to Europe via Turkey. That is how it became associated with Turkey, even though it was orginally called something else in English.
The American bird was brought to Europe later and people thought the two were related or even identical, so the new bird from the Americas became associated with Turkey too.