We present lattice QCD results for mesonic screening masses in the temperature range $140\mathrm{MeV}\lesssim T\lesssim 2500\mathrm{MeV}$. Our calculations were carried out using ($2+1$) flavors of the highly improved staggered quark action, with a physical value for the strange quark mass and two values of the light quark mass corresponding to pion masses of 160 and 140 MeV. Continuum-extrapolated results were obtained using calculations with a variety of lattice spacings corresponding to temporal lattice extents $N_{\tau }=6–16$. We discuss the implications of these results for the effective restoration of various symmetries in the high temperature phase of QCD, as well as the approach toward the perturbative limit.