The crystal lattice of Sr$_2$IrO$_4$ is investigated with synchrotron X-ray powder diffraction under hydrostatic pressures up to $P=43$ GPa and temperatures down to $20$ K. The tetragonal unit cell is maintained over the whole investigated pressure range, within our resolution and sensitivity. The $c$-axis compressibility $kappa_c(P,T) equiv -({1} / {c}) ({d c} / {d P})$ presents an anomaly with pressure at $P_1=17$ GPa at fixed $T=20$ K that is not observed at $T=300$ K, whereas $kappa_a(P,T)$ is nearly temperature-independent and shows a linear behavior with $P$. The anomaly in $kappa_c(P,T)$ is associated with the onset of long-range magnetic order, as evidenced by an analysis of the temperature-dependence of the lattice parameters at fixed $P=13.7 pm 0.5$ GPa. At fixed $T=20$ K, the tetragonal elongation $c/a(P,T)$ shows a gradual increment with pressure and a depletion above $P_2=30$ GPa that indicates an orbital transition and possibly marks the collapse of the $J_{eff}=1/2$ spin-orbit-entangled state. Our results support pressure-induced phase transitions or crossovers between electronic ground states that are sensed, and therefore can be probed, by the crystal lattice at low temperatures in this prototype spin-orbit Mott insulator.
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