  Motivated by experimental observation of the non-magnetic phase in the
compounds with frustration and disorder, we study the ground state of the
spin-$1/2$ square-lattice Heisenberg model with randomly distributed
nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ couplings. By using the
density matrix renormalization group (DMRG) calculation on cylinder system with
circumference up to $10$ lattice sites, we identify a disordered phase between
the N\'eel and stripe magnetic phase with growing $J_2 / J_1$ in the presence
of strong bond randomness. The vanished spin-freezing parameter indicates the
absence of spin glass order. The large-scale DMRG results unveil the
size-scaling behaviors of the spin-freezing parameter, the power-law decay of
the average spin correlation, and the exponential decay of the typical spin
correlation, which all agree with the corresponding behavior in the
one-dimensional random singlet (RS) state and characterize the RS nature of
this disordered phase. The DMRG simulation also provides insights and
opportunities for characterizing a class of non-magnetic states in
two-dimensional frustrated magnets with disorder. We also compare with existing
experiments and suggest more measurements for understanding the
spin-liquid-like behaviors in the double perovskite
Sr$_2$CuTe$_{1-x}$W$_{x}$O$_6$.
