Bottom-up design of biomimetic organelles has gained recent attention as a route towards understanding the transition between non-living matter and life. Despite various artificial lipid membranes being developed, the specific relations between lipid structure, composition, interfacial properties, and morphology are not currently understood. Sponge-phase droplets contain dense, nonlamellar lipid bilayer networks that capture the complexities of the endoplasmic reticulum (ER), making them ideal artificial models of such organelles. Here, we combine ultrafast two-dimensional infrared (2D IR) spectroscopy and molecular dynamics simulations to investigate the interfacial H-bond networks in sponge-phase droplets composed of glycolipid and nonionic detergents. In the sponge phase, the interfacial environments are more hydrated and water molecules confined to the nanometer-scale aqueous channels in the sponge phase exhibit dynamics that are significantly slower compared to bulk water. Surfactant configurations and microscopic phase separation play a dominant role in determining membrane curvature and slow dynamics observed in the sponge phase. The studies suggest that H-bond networks within the nanometer-scale channels are disrupted not only by confinement but also by the interactions of surfactants, which extend 1-2 nm from the bilayer surface. The results provide a molecular-level description for controlling phase and morphology in the design of synthetic lipid organelles.