The description for this achievement is some what misleading. You don't have to beat the entire story in Turbo mode. All you have to do is finish the story mission 'Super Stomp Shenanigans' with Turbo mode active. In fact you dont have to complete the entire level either. As soon as you get to the part where Dr. Von Buttnick talks about the "castle collapsing", the achievement will unlock.
Once the Story Mode has been completed you will unlock Turbo Mode, you may replay 'Super Stomp Shenanigans.' Turbo Mode makes you a lot faster, so it will take some time getting used to this mode. Turbo Mode can be switched on and off from the Main Menu.