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Developing the earlier results of Aslamazov et al (1986 Sov. Phys. JETP 64 1051) we perform a detailed study of the superconducting properties of a bilayer consisting of highly disordered superconducting (S) and low-resistive normal metal (N) films. We demonstrate theoretically and experimentally that the screening and transport properties of such a SN bilayer can be considerably enhanced compared with the single S film. Such modification originates from the proximity-induced superconductivity in the N film and it is not related to enhanced vortex pinning in the SN hybrid. In particular, the coverage of superconducting NbN and MoN films by a thin Ag or Al layer is responsible for enhancing the diamagnetic response and the critical current Ic of the SN microbridge by several times and the appearance of the diode effect in a parallel magnetic field. Due to peculiar temperature dependence of Ic and the magnetic field penetration depth, such SN bilayers can be of potential interest for designing different types of superconducting bolometers and detectors. [ABSTRACT FROM AUTHOR]