The reality is far more complicated. Propeller-driven planes remained extremely common parts of the world's air forces during the early jet era, as militaries grappled with the complexity and power of the new technology. That meant experimenting, and in the US Navy's case, marrying the two forms of flight.
Foxtrot Alpha has a comprehensive read on the FR-1 Fireball, a jet-piston fighter developed by Ryan Aeronautical, the same folks behind Charles Lindbergh's Spirit of St. Louis, during the wind down of the second World War. The XFR-1, as it was first known, was the size of a Navy F6F Hellcat. Its powertrain, though, featured a jet engine with 1,600-pounds of thrust and a Wright R-1820 Cyclone piston engine, that delivered about 1,400 horsepower.
Head over to FA for the long read on this fascinating, troubled and ultimately doomed marriage of two disparate types of propulsion.