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 ${sim}mu$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 discretisation 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 $propto Dres^{-1/2}$, 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.