Our understanding of the process through which magnetic fields reached their observed strengths in present-day galaxies remains incomplete. One of the advocated solutions is a turbulent dynamo mechanism that rapidly amplifies weak magnetic field seeds to the order of ∼|$\mu$|G. However, simulating the turbulent dynamo is a very challenging computational task due to the demanding span of spatial scales and the complexity of the required numerical methods. In particular, turbulent velocity and magnetic fields are extremely sensitive to the spatial discretization of simulated domains. To explore how refinement schemes affect galactic turbulence and amplification of magnetic fields in cosmological simulations, we compare two refinement strategies. A traditional quasi-Lagrangian adaptive mesh refinement approach focusing spatial resolution on dense regions, and a new refinement method that resolves the entire galaxy with a high resolution quasi-uniform grid. Our new refinement strategy yields much faster magnetic energy amplification than the quasi-Lagrangian method, which is also significantly greater than the adiabatic compressional estimate indicating that the extra amplification is produced through stretching of magnetic field lines. Furthermore, with our new refinement the magnetic energy growth factor scales with resolution following |$\propto {\Delta x}_\text{max}^{-1/2}$|⁠, in much better agreement with small-scale turbulent box simulations. Finally, we find evidence suggesting most magnetic amplification in our simulated galaxies occurs in the warm phase of their interstellar medium, which has a better developed turbulent field with our new refinement strategy.