We study the growth dynamics of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction-diffusion system in response to perturbations with visible light. We describe several mechanisms by which Turing patterns reappear after they are suppressed by illumination with a disc-shaped geometry. We observe that under specific conditions the patterns reorganize from a random configuration of spots and stripes to a set of ordered, concentric rings, which we refer to as target Turing patterns. These patterns closely resemble the unit cells of the Turing hexagonal superlattices known as black eye patterns. However, these target Turing patterns are not part of a larger superlattice structure, and they usually have a larger number of concentric rings. Numerical simulations support the experimental findings.