We study the dynamics of topological defects in the magnetic texture of rectangular permalloy thin-film elements during relaxation from random magnetization initial states. Our full micromagnetic simulations reveal complex defect dynamics during relaxation towards the stable Landau closure domain pattern, manifested as temporal power-law decay, with a system-size-dependent cutoff time, of various quantities. These include the energy density of the system and the number densities of the different kinds of topological defects present in the system. The related power-law exponents assume nontrivial values and are found to be different for the different defect types. The exponents are robust against a moderate increase in the Gilbert damping constant and introduction of quenched structural disorder. We discuss details of the processes allowed by conservation of the winding number of the defects, underlying their complex coarsening dynamics.