Possibly yes, turbochargers are used to make amateur turbojets. There are many out there. Search the web for sites.
The problems you will have is the quality of the shaft bushing and the provision of forced lubrication. The shape of the outer case of the rotor balances the forces on the centre bearing. You will have to design for that axial pressure.
The material used for the turbines is often different for a diesel or a gasoline engine. You will need to keep your exhaust temperature cool enough, and with the right oxygen level to prevent corrosion or melting of the turbine.