We investigate the crystal structure and lattice vibrations of Sr$_2$IrO$_4$ by a combined phonon Raman scattering and x-ray powder diffraction experiment under pressures up to $66$ GPa and room temperature. Density functional theory (DFT) and $ab$-initio lattice dynamics calculations were also carried out. A first-order structural phase transition associated with an $8$ % collapse of the $c$-axis is observed at high pressures, with phase coexistence being observed between $sim 40$ and $55$ GPa. At lower pressures, lattice and phonon anomalies were observed, reflecting crossovers between isostructural competing states. A critical pressure of $P_1=17$ GPa is associated with: (i) a reduction of lattice volume compressibility and a change of behavior of the tetragonal $c/a$ ratio take place above $P_1$; (ii) a four-fold symmetry-breaking lattice strain associated with lattice disorder; (iii) disappearance of two Raman active modes (at $sim 180$ and $sim 260$ cm$^{-1}$); and (iv) development of an asymmetric Fano lineshape for the $sim 390$ cm$^{-1}$ mode. DFT indicates that the phase above $P_1$ is most likely non-magnetic. Exploring the similarities between iridate and cuprate physics, we argue that these observations are consistent with the emergence of a rotational symmetry-breaking electronic instability at $P_1$, providing hints for the avoided metallization under pressure and supporting the hypothesis of possible competing orders that are detrimental to superconductivity in this family. Alternative scenarios for the transition at $P_1$ are also suggested and critically discussed. Additional phonon and lattice anomalies in the tetragonal phase are observed at $P_2=30$ and $P_3=40$ GPa, indicating further competing phases that are stabilized at high pressures.
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