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 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.
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