When laser beams propagate through atmospheric turbulence, the beam spreading depends on two mechanisms, i.e., free-space diffraction and atmospheric turbulence. The influence of turbulence on the beam spreading relates with diffraction. The regions spreading of partially coherent Hermite-Gaussian (H-G) beams propagating through non-Kolmogorov atmospheric turbulence are studied in detail. The propagation distance is divided into three regions, in which the beam spreading is dominated by free-space diffraction, free-space diffraction and atmospheric turbulence, and atmospheric turbulence, respectively. The influence of beam parameters and turbulence parameters on the three regions is studied, and the relation between the first region and the Rayleigh region is also examined. In addition, the physical explanations of the main results obtained are given.