In this paper, we propose and experimentally demonstrate an orthogonal subblock division multiple access (OSDMA) scheme for orthogonal frequency division multiplexing with index modulation (OFDM-IM)-based multi-user visible light communication (MU-VLC) systems, where both single-mode index modulation (SM-IM) and dual-mode index modulation (DM-IM) are considered. In order to overcome the low-pass frequency response and the light-emitting diodes (LED) nonlinearity issues of practical MU-VLC systems, OSDMA is employed together with discrete Fourier transform spreading (DFT-S) and interleaving. The feasibility and superiority of the proposed scheme have been successfully verified via both simulations and hardware experiments. More specifically, we evaluate and compare the peak-to-average power ratio (PAPR) performance and the bit error rate (BER) performance of OFDM-SM-IM, DFT-S-OFDM-SM-IM, OFDM-DM-IM and DFT-S-OFDM-DM-IM without and with interleaving. Experimental results show that remarkable distance extensions can be achieved by employing DFT spreading and interleaving for both SM-IM and DM-IM in a two-user OSDMA-VLC system.