Context. For the first time, our magnetohydrodynamical numerical calculations provide results for a three-dimensional model of barred galaxies involving a cosmic-ray driven dynamo process that depends on star formation rates. Furthermore, we argue that the cosmic-ray driven dynamo can account for a number of magnetic features in barred galaxies, such as magnetic arms observed along the gaseous arms, magnetic arms in the inter-arm regions, polarized emission that is at the strongest in the central part of the galaxy, where the bar is situated, polarized emission that forms ridges coinciding with the dust lanes along the leading edges of the bar, as well as their very strong total radio intensity. Aims. Our numerical model probes what kind of physical processes could be responsible for the magnetic field topology observed in barred galaxies (modes, etc.). We compare our modelled results directly with observations, constructing models of high-frequency (Faraday rotation-free) polarized radio emission maps out of the simulated magnetic field and cosmic ray pattern in our modeled galaxy. We also take the effects of projection into account as well as the limited resolution. Methods. We applied global 3D numerical calculations of a cosmic-ray driven dynamo in barred galaxies with different physical input parameters such as the supernova (SN) rate. Results. Our simulation results lead to the modelled magnetic field structure similar to the one observed on the radio maps of barred galaxies. Moreover, they cast new light on a number of properties in barred and spiral galaxies, such as fast exponential growth of the total magnetic energy to the present values. The quadrupole modes of magnetic field are often identified in barred galaxies, but the dipole modes (e.g., in NGC 4631) are found very seldom. In our simulations the quadrupole configuration dominates and the dipole configuration only appears once in the case of model S100, apparently as a consequence of the choice of the random number seed. Synthetic radio maps of our models display X-type structure similar to what is observed in real galaxies. Conclusions. We conclude that a cosmic-ray driven dynamo process in barred galaxies can amplify magnetic fields efficiently. The fastest rate of magnetic field increase is 195 yr for a SN frequency of 1/50 yr-1.The obtained strength of magnetic field corresponds to the observational values (a few in spiral arms). The polarization and rotation measure maps also agree with observations. We found the effect of shifting magnetic arms in 4 models (out of the sample of 5).