Based on the extended Huygens-Fresnel principle, analytical expressions for the average intensity and degree of polarization of partially coherent Bessel-Gaussian beams propagating in non-Kolmogorov turbulence were devived. The intensity distribution feature and the variation pricinple of polarization degree were studied, and the effects of the exponent parameter, structure constant, outer scale, inner scale, topological change and coherent length on the propagation properties were analyzed. The results show that the beam profile approaches to a Gaussian shape from a hollow shape and gets some spreads with increasing the value of the propagation distance. The average intensity distribution changes more quickly and gets more spreads with exponent parameter closer to 3.1, larger structure constant, larger outer scale and smaller inner scale.When the topological charge or coherent length is smaller, the average intensity distribution changes faster, but it has few effects on the spread phenomenon. Furthermore, at near distance the degree of polarization first fluctuates, then a rise and a reduce appear in succession, and when the propagation distance is long enough it tends to a stable value which equals the initial value on the source plane. The variation process is affected by the exponent parameter, structure constant, outer scale, inner scale, topological charge and coherent length.