We have measured the electric transport properties of TiN nanostrips with different widths. At zero magnetic field, the temperature-dependent resistance R(T) saturates at a finite resistance toward low temperatures, which results from quantum phase slips in the narrower strips. We find that the current-voltage (I-V) characteristics of the narrowest strips are equivalent to those of small Josephson junctions. Applying a transverse magnetic field drives the devices into a reentrant insulating phase, with I-V characteristics dual to those in the superconducting regime. The results provide evidence that our critically disordered superconducting nanostrips behave like small self-organized random Josephson networks.