  The effect of compression on the magnetic ground state of Sr$_2$IrO$_4$ is
studied with x-ray resonant techniques in the diamond anvil cell. The weak
interlayer exchange coupling between square-planar 2D IrO$_2$ layers is readily
modified upon compression, with a crossover between magnetic structures around
7 GPa mimicking the effect of an applied magnetic field at ambient pressure.
Higher pressures drive an order-disorder magnetic phase transition with no
magnetic order detected above 17-20 GPa. The persistence of strong exchange
interactions between $\mathrm{J_{eff}}=1/2$ magnetic moments within the
insulating IrO$_2$ layers up to at least 35 GPa points to a highly frustrated
magnetic state in compressed Sr$_2$IrO$_4$ opening the door for realization of
novel quantum paramagnetic phases driven by extended $5d$ orbitals with
entangled spin and orbital degrees of freedom.
