Is it possible to manufacture a jet engine that provides 500 newtons of thrust with relatively primitive tools like a lathe or so , if someone did something similar can you provide me with the knowledge you obtained please.
This would require far, far more than just a lathe. You'll have to do 3D milling of some fairly exotic materials to create turbines, sophisticated spin balancing tools, etc.
There's a DIY guy on youtube that will provide some insights on what's involved:
Also, I highly recommend AgentJayZ if you have an interest in a deeper dive into all things jet engine. He does not work on DIY engines, he overhauls/refurbs industrial and aircraft jet engines, but his videos can provide deep insights into how turbojets actually work.
To take the heat and huge rotational stresses, most of the engine has to be manufactured from superalloys which have been subject to exotic process treatments to maximize their strength. Those process steps are completely beyond the capabilities of any garage shop.
They are also enormously difficult to machine to the super high tolerances required in modern, high-powered jet engines that must last tens of thousands of hours at full power without failure.