The results of an experimental study of the abnormal domain structure kinetics in lithium niobate single crystals with a surface layer modified by soft proton exchange are presented. Domain switching in a wide field range allowed two qualitatively different types of domain structure evolution to be revealed: (1) the traditional growth of hexagonal domains in fields higher than 21.5 kV/mm and (2) the abnormal growth of stripe domains oriented along the Y crystallographic directions in the field range from 3.8 to 21.5 kV/mm. The stripe domains had a width up to 4 mu m and depth up to 30 mu m. It was shown that the time dependence of the total length of stripe domains could be analyzed in terms of the modified Kolmogorov-Avrami approach, taking into account the transition from the one-dimensional beta-model to the one-dimensional alpha-model. The possibility of the controllable creation of a quasi-periodic structure of stripe domains with an average period of 5 mu m by a two-stage polarization switching process was demonstrated. The formation and growth of stripe domains were considered in terms of the kinetic approach to the evolution of the domain structure as a result of the domain walls' motion under inefficient screening conditions caused by the presence of a modified surface layer. The abnormally low threshold fields were attributed to a presence of a "built-in" field facilitating switching, created by a composition gradient induced by soft proton exchange.