According to this site, turkeys originated in Mexico.

"About 1530, a new dish began to be put on English tables, a fowl a little larger than the traditional goose, but with a lot more meat and a refreshingly new taste. This bird had been brought to England by merchants trading out of that area of the eastern Mediterranean called the Levant but whom the English called “Turkey merchants” because that whole area was then part of the Turkish empire. The new bird was therefore called a “Turkey bird”, or “Turkey coxk”. Within a few years it had become a favourite and familiar domestic fowl, to the extent that, sixty years later, Shakespeare knew his groundlings would understand the reference to the turkey’s aggression display of blowing out its breast and strutting when he described the posturings of Malvolio in Twelfth Night:

SIR TOBY BELCH: Here’s an overwheening rogue!
FABIAN: O, peace! Contemplation makes a rare turkey-coxk of him; how he jets under his advanced plumes!

The interesting thing about the mistake over the turkey’s origins is that the English were the only people to believe they came from Turkey; nearly everyone else, including the Turks, thought they originated in India, or at least in the place they then thought was India. Turkeys actually came from Mexico and were first brought back from there about 1520, at a time when that area was called The Spanish Indies or the New Indies, illustrating the confusion in people’s minds about the true location of this new land that Columbus had found. As a result, a lot of European languages, as well as others like Arabic and Hebrew, called it something like the “bird of India”."