Recent hydrodynamical models of supernova remnants (SNRs) demonstrate that their evolution depends heavily on the inhomogeneities of the surrounding medium. As SNRs expand, their morphologies are influenced by the nonuniform and turbulent structure of their environments, as reflected in their radio continuum emission. In this paper, we measure the asymmetries of 96 SNRs in radio continuum images from three surveys of the Galactic plane and compare these results to the SNRs’ radii, which we use as a proxy for their age. We find that larger (older) SNRs are more elliptical/elongated and more mirror asymmetric than smaller (younger) SNRs, though the latter vary in their degrees of asymmetry. This result suggests that SNR shells become more asymmetric as they sweep up the interstellar medium (ISM), as predicted in hydrodynamical models of SNRs expanding in a multiphase or turbulent ISM.