In the first two places in those sentences, "the turkey" refers to the large bird. The last one could refer to either, but is probably referring to the meat.
In the sentence you provided, the word "turkey" refers to both a large bird and the meat from a turkey. When we say "we cooked a big turkey," it implies that they cooked a whole turkey, which is a large bird. Then, when we say "the turkey was finally ready," it indicates that the meat from the cooked turkey was ready to be served.
