Swapping out your stock L-series engine with something more powerful is better than just hitting the turbo.
As to what engines are optimal for Fit models, there are the D16 or D17 engines used for Integras and Civics. You can also find the engines of the Honda B-series. These are the most common engines that have been switched for Honda Fit, and each of them has a lot of aftermarket performance parts available.
However, to make the most of your swap, it's advisable to get sport headers, a larger radiator, and an improved exhaust.
The larger radiator will have more optimal cooling while the better headers will boost exhaust and intake gas flow. Eventually, a strong exhaust will provide the engine with a quicker way to expel exhaust gases.
If you haven't swapped your Fit's L-series engine, for the 2004-2007 Fit, we suggest HotPipes Headers HP-HND00 for headers. On the other hand, the HotPipes HP-HND006 is suitable for the 2008-2013 Fit, while the HND008 is good for the 2014-2020 Fit.