In relativistic magnetized plasmas, asymmetry in the number densities of left-and right-handed fermions, i.e., a nonzero chiral chemical potential mu(5), leads to an electric current along the magnetic field. This causes a chiral dynamo instability for a uniform mu(5), but our simulations reveal a dynamo even for fluctuating mu(5) with zero mean. It produces magnetically dominated turbulence and generates mean magnetic fields via the magnetic alpha effect. Eventually, a universal scale-invariant k(-1) spectrum of mu(5) and a k(-3) magnetic spectrum are formed independently of the initial condition.