We present some results obtained by numerical modeling of the propagation of vortex beams with a spiral phase through a randomly inhomogeneous medium being presented by a phase screen placed in the beginning of the propagation path. Such beams, if propagated under conditions of weak turbulence, also experience distortions, like Gaussian beams. However, the statistically averaged vortex beams conserve the central intensity dip with a nonzero intensity on the beam axis. The greater the beam vortex charge, the longer the beam propagation distance, at which the central dip is not smeared. The vortex beams being the Laguerre-Gaussian modes are found to have the same broadening properties while propagated through a randomly inhomogeneous medium as the Gaussian beams. The broadening of averaged vortex beams does not depend on the vortex charge and coincides with the broadening of a Gaussian beam.