The lateral movement in lipid membranes depends on their diffusion constant within the membrane. However, when the flux of the subphase is high, the convective flow beneath the membrane also influences lipid movement. Lipid monolayers of an unsaturated fatty acid at the water-air interface serve as model membranes. The formation of domains in the liquid/condensed coexistence region is investigated. The dimension of the domains is fractal, and they grow with a constant growth velocity. Increasing the compression speed of the monolayer induces a transition from seaweed growth to dendritic growth. Seaweed domains have broad tips and wide and variable side branch spacing. In contrast, dendritic domains have a higher fractal dimension, narrower tips, and small, well-defined side branch spacing. Additionally, the growth velocity is markedly larger for dendritic than seaweed growth. The domains' growth velocity increases and the tip radius decreases with increasing supersaturation in the liquid/condensed coexistence region. Implications for membranes are discussed.