I imagine it would be a total fail. If jet engine produced linear acceleration is greater than internal combustion engine linear acceleration,- then there will be unavoidable tire slipping, due to the fact that car's engine is not on par with jet engine, hence in this scenario car's engine would act as passive breaks.
Second point - maneuverability issues. While car's torque in case of driver tilted wheels nicely rotates car in a turn,- jet engine would have just pushed car simply forward in old direction, resulting again in tires slipping, because jet force is applied to car COM. Have you seen an aircrafts rushing in turns at airports at $50~km/h$ ? They are very slow at this. This could be a bit fixed by rotating jet engine in a turn,- for producing acceleration in a direction which driver wants to be. But this have to be strictly synchronized with turning of wheels. Which is again a big issue.
Lastly as you mentioned already, jet engine has poor response,- it's good for long-lasting acceleration, but not for sudden and frequent acceleration changes to which internal combustion engine is optimized for. If jet engine again will be not synchronized with a car engine by responsiveness,- it will result in tire slipping, maneuverability issues, car engine fuel consumption ineffectiveness and/or other problems.
Well... unless you want to beat Guinness World Record on fastest car in a world,- then remove car engine at all, attach jet engine to it, and accelerate only in a straight line, then you will beat them all. But your mixed solution for maneuvering in a cities where we spend at least $20\%$ of time just standing in traffic jams,- is a no-go.