We present an investigation of the effect of randomizing exchange strengths in the $S=1/2$ square lattice quasi-two-dimensional quantum Heisenberg antiferromagnet (QuinH)$_2$Cu(Cl$_{x}$Br$_{1-x}$)$_{4}cdot$2H$_2$O (QuinH$=$Quinolinium, C$_9$H$_8$N$^+$), with $0leq x leq 1$. Pulsed-field magnetization measurements allow us to estimate an effective in-plane exchange strength $J$ in a regime where exchange fosters short-range order, while the temperature $T_{mathrm{N}}$ at which long range order (LRO) occurs is found using muon-spin relaxation, allowing us to construct a phase diagram for the series. We evaluate the effectiveness of disorder in suppressing $T_{mathrm{N}}$ and the ordered moment size and find an extended disordered phase in the region $0.4 lesssim x lesssim 0.8$ where no magnetic order occurs, driven by quantum effects of the exchange randomness.
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