Laser-induced damage of fused silica optics at 351 nm is a key factor limiting the output energy of high-power laser facility, especially the damage growth process. A comprehensive understanding of its damage growth behavior is of critical importance for high-power laser facility. Thus we study the laser-induced damage growth on the exit surface of fused silica under the subsequent illumination of 5 ns square pulses at 351 nm on a large-aperture high-power laser facility. Experiment is conducted with a 36 cm thick UV grade fused silica focus lens in clean atmosphere and at room temperature. 56 laser shots of 3 fluence in a range from 0.1 J/cm2 to 8.1 J/cm2 are fired during the experiment. And the damage initiation process and growth process are monitored and recorded with an online optics damage inspection instrument which has an optical resolution of about 50 m. Experimental results demonstrate that the sizes of exit-surface damage sites exponentially or linearly grow with laser shots and the damage growth rate increases with laser fluence. However, it is found that even under the same laser conditions the damage grow rate is not a fixed value, which means that besides the laser fluence other parameters also influence the damage grow process. In order to highlight some tendencies, we consider the single-shot damage growth rate and calculate the average of inside fluence bins. Statistical analysis shows that smaller sites tend to grow with larger growth rates than larger sites under the irradiation of the same laser fluence. This result indicates that damage growth rate is influenced by both laser fluence and damage site size. It suggests that the damage growth rule needs to be incorporated into a size-dependent growth effect. The result that higher growth rates are obtained for small damage sites may be related to the damage growth mechanism of fused silica. Damage crater of fused silica consists of a central core and numerous surrounding cracks. The defects in the central core absorb laser energy and yield plasma, then the plasma pressure will open the cracks on the periphery of the crater and lead to lateral and axial expansion of cracks which can be identified as damage growth. The fact that smaller sites grow faster than larger sites implies that smaller sites more efficiently couple laser energy into fracture energy. Our results have important implications for both the prediction of fused silica optics lifetime and the fundamental understanding of laser damage mechanism.