The analytical expressions for the spectral degree of coherence, the effective radius of curvature and the propagation factor of the Bessel-Gaussian Schell-model (BGSM) beam in turbulent atmosphere are derived based on the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function (WDF). The evolution properties of BGSM beams propagating in non-Kolmogorov turbulence are investigated by a set of numerical examples. It is demonstrated that the spectral degree of coherence of the BGSM beam evolves into Gaussian profile twice with the increasing of the propagation distance. The turbulence-induced degradation can be remarkably reduced by using the BGSM beam with the proper source parameters. The effects that the generalized refractive-index structure constant, outer and inner scales, and the spectral index of spatial power spectrum of atmospheric turbulence have on the evolution properties of BGSM beams are also discussed in detail.