The average intensity of the Bessel Gaussian beams propagating through the non-Kolmogorov turbulence based on Rytov theory is derived without the quantic approximation in this paper. Therefore, this result is comparatively more accurate than that calculated by the extended Huygens-Fresnel principle, especially when the inner scale of the turbulence is small or the beams width is large. There is an interesting finding which does not exist in Gaussian beams propagation. It is the intensity variation with the inner scale that displays different behaviors when the beams width is different. Moreover, there will be some beams with specific source width, whose average intensities on the axis do not affected by the turbulence after the inner scale increasing to a certain value as their turbulence perturbation is zero. And the beams here become to the flat top beams. In summary, this paper provides an accurate method for the investigation of the Bessel Gaussian beams propagation through the non-Kolmogorov turbulence and improves the theoretical basis for the applications.