The promising potential of underwater applications in visible light communication (VLC) systems has recently gained considerable research attention, as an efficient technology for enabling high data rate, massive connectivity, and ultra-low latency. Different from indoor VLC, underwater wireless communications experience harsh environmental challenges, yielding a degraded performance. Non-orthogonal multiple access (NOMA) was introduced to enhance spectral efficiency and connectivity of underwater VLC communications. In this paper, we develop a mathematical framework to evaluate the performance of NOMA-enabled underwater VLC systems in the presence of turbulence. Specifically, we derive a closed-form expression for the outage probability of NOMA over lognormal channels, while considering the effect of path loss and turbulence. The derived analytical results with the corresponding numerical results demonstrate that the transmission distance between laser diode and sensor nodes (SNs) has a high impact on the outage performance of the SNs, due to the increased turbulence level with increased distance. Moreover, the presented results show the effect of the water type on the outage performance of the SNs and on the optimum values of the power coefficients.