Context. Dynamical motions in the intra-cluster medium (ICM) can imprint distinctive features on X-ray images that map the thermal bremsstrahlung emission from galaxy clusters, such as sharp surface brightness discontinuities due to shocks and cold fronts. The gas dynamics during cluster mergers may also drive large-scale turbulence in the ICM, which in turn generates extended (megaparsec-scale) synchrontron sources known as radio halos. Aims. Surface brightness edges have been found numerous times in the thermal gas of clusters based on X-ray observations. In contrast, edges in radio halos have only been observed in a handful of cases. Our goal is to search for new radio surface brightness discontinuities in the ICM. Methods. We inspected the images of the Bullet Cluster and the other 25 radio halos reported in the MeerKAT Galaxy Cluster Legacy Survey. To aid the identification of surface brightness discontinuities, we applied a gradient-filtering edge-detection method to the radio images. Results. We find that the adopted filtering technique is helpful in identifying surface brightness edges in radio images, allowing us to identify at least one gradient in half of the radio halos studied. For the Bullet Cluster, we find excellent agreement between the locations of the four radio discontinuities detected and the X-ray edges. This similarity informs us that there is substantial interplay between thermal and nonthermal components in galaxy clusters. This interplay is likely due to the frozen-in ICM magnetic field, which mediates the advection of cosmic rays while being dragged by thermal gas flows. Conclusions. We conclude that radio halos are shaped by dynamical motions in the ICM and that they often display surface brightness discontinuities, which appear to be co-located with edges in the thermal gas emission. Our results demonstrate that new and future generations of radio telescopes will provide an approach to efficiently detecting shocks and cold fronts in the ICM that is complementary to X-rays.