Vortex beams carrying orbital angular momentum (OAM), which feature helical wavefronts, have been regarded as an alternative degree of freedom for free-space optical (FSO) communication systems. However, in practical applications, atmospheric turbulence and limited-size receiving aperture effects will cause OAM modal degradation and seriously reduce the received power. In this paper, by controlling the radial phase distribution of conventional OAM beams, quasi-ring Airy vortex beams (QRAVBs) are successfully generated in the experiments to increase the received power under the limited-size receiving aperture conditions. By employing 72-Gbit/s 16-ary quadrature amplitude modulation (16-QAM) discrete multi-tone (DMT) signals, we successfully demonstrate free-space data transmission with QRAVBs in the experiments. Moreover, the transmission performance of QRAVBs under atmospheric turbulence is also evaluated. Comparing with conventional OAM beam and Bessel beam, the obtained results show that QRAVBs can achieve higher received power and better BER performance under limited-size receiving aperture and atmospheric turbulence conditions.