In free-space optical (FSO) communications, the orbital angular momentum (OAM) multiplexing/demultiplexing of Bessel beams perturbed by atmospheric turbulence is of great significance. We used the Gerchberg-Saxton algorithm without a beacon beam to compensate for the aberrant helical phase of the Bessel beam distorted by the turbulent atmosphere. The optical vortex Dammann axicon grating was applied for the simultaneous measurement of the intensities of the demodulated spectra of the OAM modes of the Bessel beams disturbed by atmospheric turbulence. The experimental results demonstrate that the distorted phase of the Bessel beam can be compensated and the mode purity of the target OAM mode is enhanced from 0.85 to 0.92 in case of weak turbulence. Our results will improve the quality of the OAM modes of Bessel beam (de)multiplexing in FSO communication systems.