A numerical study of magnetosheath ion motion in the magnetopause current sheet in the presence of magnetic fluctuations is reported. Test particles are injected at the magnetosheath side only. The magnetic field turbulence is modeled as a power law spectrum in phase space, which reaches maximum intensity in the center of the magnetopause current sheet and decreases toward the magnetosheath and magnetosphere boundaries, creating magnetic islands. The number of particles entering the magnetosphere, reflected from the magnetopause, and flowing away from the flanks is computed, as a function of the fluctuation level of the turbulence and of the magnetic field shear parameter. All those quantities appear to be strongly dependent on the fluctuation level, with the number of particles entering the magnetopause increasing with the fluctuation level. The density profile for a localized source in the magnetosheath is not Gaussian but rather has a long tail extending into the magnetopause. The statistical analysis of the ion motion along the direction normal to the magnetopause reveals that the ions perform a L¿vy random walk, so that the penetrating particles could reach the magnetospheric side of the simulation box with few long jumps in between the magnetic islands. This shows that plasma transport across the magnetopause in presence of magnetic turbulence is non-Gaussian and needs to be described in terms of enhanced, superdiffusive transport.