Cells generally convert external nutrient resources to support metabolism and growth. Understanding the thermodynamic efficiency of this conversion is essential to determine the general characteristics of cellular growth. Using a simple protocell model with catalytic reaction dynamics to synthesize the necessary enzyme and membrane components from nutrients, the entropy production per unit-cell-volume growth is calculated analytically and numerically based on the rate equation for chemical kinetics and linear nonequilibrium thermodynamics. The minimal entropy production per unit-cell growth is found to be achieved at a nonzero nutrient uptake rate rather than at a quasistatic limit as in the standard Carnot engine. This difference appears because the equilibration mediated by the enzyme exists only within cells that grow through enzyme and membrane synthesis. Optimal nutrient uptake is also confirmed by protocell models with many chemical components synthesized through a catalytic reaction network. The possible relevance of the identified optimal uptake to optimal yield for cellular growth is also discussed.