<p>In semi-arid areas, grazing intensity has a certain impact on vegetation, and the non-local interaction of roots in absorbing water also plays an indispensable role in vegetation. So in this paper, a vegetation-water model considering grazing intensity and non-local delay is established. The conditions for Turing patterns occurrence in the vegetation-water model are determined through an analytical method. Additionally, the multiple-scale analysis is employed to get the amplitude equations at the critical value of Turing bifurcation. Through the stability analysis of amplitude equations, we determine the conditions for the emergence of different Turing patterns. Various spatial distributions of vegetation in the semi-arid areas are qualitatively described by numerical simulations. The intensification of grazing intensity will change the structural arrangement of vegetation patterns: spot patterns → mixed patterns → stripe patterns. The results indicate that the overall trend of vegetation growth would be better with the increase of grazing intensity. Besides, we demonstrate that the non-local interaction between vegetation and water plays a key role in the formation of vegetation patterns, offering a theoretical foundation for the conservation and restoration of vegetation.</p>