Because of their weak interlayer bonding, van der Waals (vdW) solids are very sensitive to external stimuli such as strain. Experimental studies of strain tuning of thermal properties in vdW solids have not yet been reported. Under ∼9% cross-plane compressive strain created by hydrostatic pressure in a diamond anvil cell, we observed an increase of cross-plane thermal conductivity in bulk MoS_{2} from 3.5 to about 25 W m^{-1} K^{-1}, measured with a picosecond transient thermoreflectance technique. First-principles calculations and coherent phonon spectroscopy experiments reveal that this drastic change arises from the strain-enhanced interlayer interaction, heavily modified phonon dispersions, and decrease in phonon lifetimes due to the unbundling effect along the cross-plane direction. The contribution from the change of electronic thermal conductivity is negligible. Our results suggest possible parallel tuning of structural, thermal, and electrical properties of vdW solids with strain in multiphysics devices.