Osmotic pressure influences cellular shape. In a growing cell, chemical reactions and dilution induce changes in osmolarity, which in turn influence the cellular shape. Using a protocell model relying upon random conservative chemical reaction networks with arbitrary stoichiometry, we find that when the membrane is so flexible that its shape adjusts itself quasi-instantaneously to balance the osmotic pressure, the protocell either grows filamentous or fails to grow. This behavior is consistent with a mathematical proof. This suggests that filamentation may be a primitive growth mode resulting from the simple physical property of balanced osmotic pressure. We also find that growth is favored if some chemical species are only present inside the protocell, but not in the outside growth medium. Such an insulation requires specific chemical schemes. Modern evolved cells such as E. coli meet these requirements through active transport mechanisms such as the phosphotransferase system.