A new mechanism of internal wave breaking in the subsurface ocean layer is considered. The breaking is due to the ‘resonant’ interaction of shoaling long internal gravity waves with the subsurface shear current occurring in a resonance zone. Provided the wind-induced shear current is oriented onshore, there exists a wide resonance zone, where internal wave celerity is close to the current velocity at the water surface and a particularly strong resonant interaction of shoaling internal waves with the current takes place. A model to describe the coupled dynamics of the current perturbations treated as ‘vorticity waves’ and internal waves propagating over a sloping bottom is derived by asymptotic methods. The model generalizes the earlier one by Voronovich, Pelinovsky & Shrira (1998) by taking into account the mild bottom slope typical of the oceanic shelf. The focus of the work is upon the effects on wave evolution due to the presence of the bottom slope. If the bottom is flat, the model admits a set of stationary solutions, both periodic and of solitary wave type, their amplitude being limited from above. The limiting waves are sharp crested. Space–time evolution of the waves propagating over a sloping bottom is studied both by the adiabatic Whitham method for comparatively mild slopes and numerically for an arbitrary one. The principal result is that all onshore propagating waves, however small their initial amplitudes are, inevitably reach the limiting amplitude within the resonance zone and break. From the mathematical viewpoint the unique peculiarity of the problem lies in the fact that the wave evolution remains weakly nonlinear up to breaking. To address the situations when the subsurface current becomes strongly turbulent due to particularly intense wind-wave breaking, the effect of turbulent viscosity on the wave evolution is also investigated. The damping due to the turbulence results in a threshold in the initial amplitudes of perturbations: the ‘subcritical’ perturbations are damped, the ‘supercritical’ ones inevitably break. As the breaking events occur mainly in the subsurface layer, they may contribute significantly to the mixing and exchange processes at the air/sea interface and in creating significant surface signatures.