We report on magnetization M(H), dc and ac magnetic susceptibility χ(T), specific heat C_{m}(T) and muon spin relaxation (μSR) measurements of the Kitaev honeycomb iridate Cu_{2}IrO_{3} with quenched disorder. In spite of the chemical disorders, we find no indication of spin glass down to 260 mK from the C_{m}(T) and μSR data. Furthermore, a persistent spin dynamics observed by the zero-field muon spin relaxation evidences an absence of static magnetism. The remarkable observation is a scaling relation of χ[H,T] and M[H,T] in H/T with the scaling exponent α=0.26-0.28, expected from bond randomness. However, C_{m}[H,T]/T disobeys the predicted universal scaling law, pointing towards the presence of additional low-lying excitations on the background of bond-disordered spin liquid. Our results signify a many-faceted impact of quenched disorder in a Kitaev spin system due to its peculiar bond character.