A defining characteristic of silicate melts is the degree of polymerization (tetrahedral connectivity), which dictates viscosity and affects compressibility. While viscosity of depolymerized silicate melts increases with pressure consistent with the free-volume theory, isothermal viscosity of polymerized melts decreases with pressure up to ~3-5 GPa, above which it turns over to normal (positive) pressure dependence. Here we show that the viscosity turnover in polymerized liquids corresponds to the tetrahedral packing limit, below which the structure is compressed through tightening of the inter-tetrahedral bond angle, resulting in high compressibility, continual breakup of tetrahedral connectivity and viscosity decrease with increasing pressure. Above the turnover pressure, silicon and aluminium coordination increases to allow further packing, with increasing viscosity and density. These structural responses prescribe the distribution of melt viscosity and density with depth and play an important role in magma transport in terrestrial planetary interiors.