As said the F-5 was originally procured as a light fighter to sell/give to friendly countries. It's J85 engines were actually originally designed for the ADM20 Quail B-52 decoy missile. These engines also powered some learjet models and the white knight carrier aircraft for the recent SpaceShipOne suborbital vehicle and has the highest thrust to weight ratio of any production jet engine available.
The F20 was something of an F-16 competitor, but it was upgraded from the basic F-5, with proper avionics and a single F404 engine as used in the F/A-18 (an excellent engine and state of the art at the time). It was an excellent aircraft, with a low radar cross section and excellent power to weight ratio. It had excellent climb rate and low maintainence, it could be ready for combat within just 1 minute. It's main flaws were a lack of horizontal stability in some areas of the envelope but probably the biggest flaw was that the wing loading was far too high due to a design error and/or because they did it quickly and on the cheap, not bothering to change the F-5 wing much. This meant it could barely outturn an F-4 Phantom and by the time they redesigned it, most prospective countries had gone for the F-16. The fact that none of the US services wanted to buy any, not even as agressors pretty much sealed it's fate. The final nail in the coffin was that two of the three demo aircraft crashed while on demonstration flights, not a good advert. Apparently the last is still flying/flyable in a museum somewhere.
The F-20, like that other slightly later Northrop marketing failure, The F-18L, was an aircraft that could have been an excellent machine and would have eclipsed it's rivals in most areas but failed mainly due to not having a US order and poor/rival marketing.