Whilst many metal-organic frameworks possess the chemical stability needed to be used as functional materials, they often lack the physical strength required for industrial applications. Herein, we have investigated the mechanical properties of two UiO-topology Zr-MOFs, the planar UiO-67 ([Zr6O4(OH) 4(bpdc) 6], bpdc: 4,4'-biphenyl dicarboxylate) and UiO-abdc ([Zr6O4(OH) 4(abdc) 6], abdc: 4,4'-azobenzene dicarboxylate) by single-crystal nanoindentation, high-pressure X-ray diffraction, density functional theory calculations, and first-principles molecular dynamics. On increasing pressure, both UiO-67 and UiO-abdc were found to be incompressible when filled with methanol molecules within a diamond anvil cell. Stabilization in both cases is attributed to dynamical linker disorder. The diazo-linker of UiO-abdc possesses local site disorder, which, in conjunction with its longer nature, also decreases the capacity of the framework to compress and stabilizes it against direct compression, compared to UiO-67, characterized by a large elastic modulus. The use of non-linear linkers in the synthesis of UiO-MOFs therefore creates MOFs that have more rigid mechanical properties over a larger pressure range.