We have addressed numerically and analytically the diffusion mechanism of Turing pattern formation for the chlorine-iodine-malonic-acid system with a local concentration depended diffusivity (LCDD) in a two-dimensional space as well as its reciprocal counterpart. It is found that the Turing pattern develops in a normal way of Fickian diffusion when LCDD is not involved, namely with a zero LCDD adjusting parameter. The formation of a Turing pattern evolves in an abnormal way of either subdiffusion or superdiffusion when a non-zero LCDD adjusting parameter is taken into account, and a negative or a positive value of LCDD parameter corresponds to a subdiffusion or a superdiffusion. The sensitivity of the steady amplitude of the activator concentration of a Turing system to a random initial condition increases with increasing LCDD parameter.