The similarities to cuprates make iridates an interesting potential platform for investigating superconductivity. Equally attractive are their puzzling complex intrinsic interactions. Here, we report an ultrafast optical spectroscopy investigation of a coherent phonon mode in Sr3Ir2O7, a bilayer Ruddlesden-Popper perovskite iridate. An anomaly in the A1g optical phonon (ν=4.4THz) is unambiguously observed below the Néel temperature (TN), which we attribute to pseudospin-lattice coupling (PLC). Significantly, we find that PLC is the dominant interaction at low temperature, and we directly measure the PLC coefficient to be λ=150±20cm-1, which is two orders of magnitude higher than that in manganites (<2.4cm-1) and comparable to that in CuO (50cm-1, the strongest PLC or spin-lattice coupling (SLC) previously known). Moreover, we find that the strong PLC induces an anisotropic negative thermal expansion. Our findings highlight the key role of PLC in iridates and uncovers another intriguing similarity to cuprates.