The single-layered Ruddlesden-Popper iridate Sr₂IrO₄ hosts a J_eff =1/2 state and an ab-plane canted antiferromagnetic structure, making it a promising candidate for hosting a range of exotic states. Pseudospins arising from the strong spin-orbit coupling (SOC) at the Ir sites have non-negligible coupling with the lattice and may be responsible for a series of finely-balanced magnetic transitions. In particular, we report a pressure-driven spin-flip transition to c-axis aligned antiferromagnetic order at ~10 GPa under hydrostatic conditions, using optical Raman scattering measurements to reveal a concurrent first-order structural transition. The phonon anomalies we observe through temperature- and polarization-dependent measurements indicate that the new phase embraces strong phonon anharmonicity and/or enhanced directional negative thermal expansion. The findings underscore the delicate interplay between the SOC and lattice in 5d materials and provide insights for potential functionalities.