Organic glasses can grow crystals much faster on the free surface than in the interior, a phenomenon important for fabricating stable amorphous materials. This surface process differs from and is faster than the glass-to-crystal (GC) growth mode existing in the bulk of molecular glasses. We report that similar to GC growth, surface crystal growth terminates if glasses are heated to gain fluidity. In their steady growth below the glass transition temperature T-g, surface crystals rise above the amorphous surface while spreading laterally and are surrounded by depressed grooves. Above T-g, the growth becomes slower, sometimes unstable. This damage is stronger on segregated needles (alpha indomethacin, nifedipine, and o-terphenyl) than on crystals growing in compact domains (gamma indomethacin). This effect arises because the onset of liquid flow causes the wetting and embedding of upward-growing surface crystals. Segregated needles are at greater risk because their slow-growing flanks appear stationary relative to liquid flow at a low temperature. The disruption of surface crystal growth by fluidity supports the view that the process occurs by surface diffusion, not viscous flow. Compared to the bulk GC mode, surface crystal growth is disrupted less abruptly by fluidity. Nevertheless, to the extent that fluidity damages them, both processes are solid-state phenomena terminated in the liquid state.