Context. The galactic winds of starburst galaxies (SBGs) give rise to remarkable structures on kiloparsec scales. However, the evolution and shape of these giant wind bubbles, as well as the properties of the shocks they develop, are not yet fully understood. Aims. We aim to understand what shapes the galactic winds of SBGs, with a particular focus on the role of large-scale magnetic fields in the dynamical evolution of galactic wind-inflated bubbles. In addition, we aim to explore where the conditions for efficient particle acceleration are met in these systems. Methods. We performed magnetohydrodynamic simulations with the AMRVAC code (Adaptive Mesh Refinement Versatile Advection Code) with various configurations of the galactic medium density profile and magnetization. Results. We observe that the large-scale magnetic field, in which galactic winds expand, can impact the structure and evolution of inflated bubbles. However, the typical structures observed in starburst galaxies, such as M82, cannot be solely explained by the magnetic field structures that have been considered. This highlights the importance of other factors, such as the galactic disk, in shaping the galactic bubble. Furthermore, in all the magnetized cases we investigated, the forward wave resulting from the expanding bubbles only results in compression waves, whereas the wind termination shock features high Mach numbers, making it a promising site for diffusive shock acceleration up to a ¼102a PeV. The synthetic X-ray images generated from our models reveal an envelope surrounding the bubbles that extends up to 2 kpc, which could correspond to the polarized emission observed from planar geometry in M82, as well as a large structure inside the bubble corresponding to the shocked galactic wind. Additionally, our findings indicate that, as observed with the SOFIA instrument, a large ordered magnetic field is associated with the free galactic wind, while a more turbulent magnetic field is present in the shocked region.