Sorry, when I wrote ...
We don't want things looking like this do we 
Just kidding.
... I forgot to post the image, because initially I couldn't find it.
What Warpspeed wrote sounds right and he clearly knows more about this than I do, but I question whether it applies or distracts.
A link was given (
to a product which reportedly increases horsepower.
Consideration will have been made in the design of this product of whether gas can come out of one cylinder and end up trying to get back into another. A net horsepower gain says that things aren't too bad, athough we don't know how it operates over the whole range, but presumably no worse if the product avoids customer complaints.
So as I see it, the real question is not the fundamentals of its design, but whether adding a turbo after it would cause significant changes to the advantages we assume it already has.
The pipes seem to merge quite close to the end.
For gas flow from one tube to undesirably interfere
more (than in the non turbo version) with flow in another in the way Warpspeed described, I think flow speed (m/s) would have to be higher with the turbo.
With the turbo, there will be more mass of gas. When in the cylinders, its at a higher density and higher pressure. But does it come out faster? Or just at a similar speed maintaining high pressure?
(1)
The higher pressure relative to the outside world (atmospheric pressure) is a reason the gas could speed up
more compared to a normally aspirated vehicle.
(2)
The blockage of the turbo is a reason the gas could speed up
less compared to a normally aspirated vehicle.
Again, simply a gut feeling ...
the restriction of the turbo would cause high pressures to be maintained until after the turbo, before the turbo there wouldn't be an increase in the speed (m/s) of gas flow compared to the normally aspirated version with this same "shorty header" (so we wouldn't have the problems warpspeed descibes). What I'm saying, is that effect 2 - as a guess - overrides effect 1 in this case.
Taking some merely illustrative figures ...
If the "shorty header" alone gives you 2% more horsepower,
and the turbo alone gives you 50% more, then
1.5*1.02=1.53
then I suspect you might get a touch more than 53% more power with the two combined.
==
I've changed my tune a little since the first posting since I have cottoned-on that the "shorty header" provides a power increase in its intended application, probably with a mixture of warpspeed's understanding and pat's experience tempered calculations.