We present a comprehensive study of the effect of hydrostatic pressure on the magnetic structure and spin dynamics in the spin-1/2 coupled ladder compound C$_9$H$_{18}$N$_2$CuBr$_4$. The applied pressure is demonstrated as a parameter to effectively tune the exchange interactions in the spin Hamiltonian without inducing a structural transition. The single-crystal heat capacity and neutron diffraction measurements reveal that the N$rm acute{e}$el ordered state breaks down at and above a critical pressure $P_{rm c}$$sim$1.0 GPa through a continuous quantum phase transition. The thorough analysis of the critical exponents indicates that such transition with a large anomalous exponent $eta$ into a quantum-disordered state cannot be described by the classic Landaus paradigm. Using inelastic neutron scattering and quantum Monte Carlo methods, the high-pressure regime is proposed as a $Z_2$ quantum spin liquid phase in terms of characteristic fully gapped vison-like and fractionalized excitations in distinct scattering channels.
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