This post on the GT5 forums from Jan 2012 was spot on...

[ML]

Back to GT86!

There are no turbo/compressor because both Toyota and Subaru are focusing on this being a NA car. Both manufacturers have said there will be no turbo on this car. Even the STI version will not have turbo. (Subaru say)

They might say that just for marketing reasons, not wanting people to wait for a turbo version and not buy the car in the beginning. But I really think this car will focus on being NA.

Fitting a turbo on this engine in the GT86/BRZ chassis is not as easy due to its the engines low mounting position, and the available space under the hood. They say if the car would have a turbo, the car would have been engineered with the space for it, and a drivetrain suitable for the added HP and torque. Making it heavier, and more expensive.
The same engine block with turbo will be in the new Impreza WRX. But it is mounted much higher there. And maybe with a drivetrain more suitable.
(When that said. I do not think the GT86s drivetrain is weak. It think it will handle a lot more power. We already know transmission and diff is from/based on higher powered lexus if i remember correctly.)

Also, I think a BRZ with turbo would mess too much up the the impreza line up. The WRX (STI) would not like to have a lower range car be faster.
Scion FR-S, the US version of the car is aimed at younger people. Generally cheaper cars. A high performance, high price car car will not fit the Scion brand.
The Toyota GT86 is based on three cars. But it is mainly a tribute to the AE86 models. The focus was affordable, and fun without extreme power, turbo (lag), or super sticky tires.
Most of the tuners have also been focusing on this car being light weight with a high reving NA engine (talking about the old 86s)
This might also have something to do with the high compression on the engine.

We will probably see aftermarket turbo or supercharger kits in the future, because there will always be people wanting it.
But the normal road car will not focus on having a turbo, so neither did GT5.
I think Toyota had something to say about that too. Im pretty sure they used GT5 as a marketing channel. It had own promo videos when being released. They have had several concepts of the car in GT5. It is the ONLY FREE road car that has been released to the game. And for some reason, it has sport hard as standard. Making it seem faster and gripper right out from the dealership compared to its rivals.
It think all that is marketing. And not having a turbo option in the game would make scene if Toyota have no plans of fitting a turbo in real life life.

The car will max out right above the 500pp range. That is in my opinion very good for a NA car that originally had 198bhp.

http://www.gtplanet.net/forum/showthread.php?t=239245

I tried to find out why your couldn't get a turbo or a supercharger in GT5 for the twins, and this old post seemed to be spot on with what has happened, except for the part about not being engineered for a turbo.