In Network Dataset, Global Turn Delay is in seconds. Whereas my cost attribute is in minutes. Also my turn feature class impedance is in minutes. Is there any need to convert turn impedance to seconds and make another cost attribute having units as "seconds"?
Absolutely not. You can define your drivetime cost in minutes or any other unit, having global turns in seconds at the same time. It will convert your units on-the-fly correctly. Just to prove the point, you can change the global turn value for left turns and see how the value in seconds will be converted to minutes in the Directions window (and of course taken into consideration when solving the route).
Here I've set my left turns to be 1000 seconds and you can see that it was converted to minutes with no problem.