The chicken would win the race, after 21.6 minutes.
If you calculate the weighted average, you'll see that the chicken's average speed, after 5 minutes, is 2.6 mph. The turkey's average speed is 2.666666... mph. The crow, of course, has an average speed of 2 mph. From these figures, you might guess that the turkey would win.
The catch is the fact that the one-mile race does not take an even multiple of five minutes to complete. The chicken's average speed is only accurate after the full four minutes of walking is completed: at any time before that, the chicken's mean speed will be higher.
After 20 minutes have passed, the chicken will have travelled 0.86666.... miles, and the turkey will have travelled 0.888888... miles. However, the chicken will then begin flying at 5 mph, and will overtake the turkey. The final results will look like this:
Chicken: 21.6 minutes.
Turkey: 22.5 minutes.
Crow: 30 minutes.