(abridged) We report rest-frame submillimeter H2O emission line observations of 11 HyLIRGs/ULIRGs at z~2-4 selected among the brightest lensed galaxies discovered in the Herschel-ATLAS. Using the IRAM NOEMA, we have detected 14 new H2O emission lines. The apparent luminosities of the H2O emission lines are $mu L_{rm{H_2O}} sim 6-21 times 10^8 L_odot$, with velocity-integrated line fluxes ranging from 4-15 Jy km s$^{-1}$. We have also observed CO emission lines using EMIR on the IRAM 30m telescope in seven sources. The velocity widths for CO and H2O lines are found to be similar. With almost comparable integrated flux densities to those of the high-J CO line, H2O is found to be among the strongest molecular emitters in high-z Hy/ULIRGs. We also confirm our previously found correlation between luminosity of H2O ($L_{rm{H_2O}}$) and infrared ($L_{rm{IR}}$) that $L_{rm{H_2O}} sim L_{rm{IR}}^{1.1-1.2}$, with our new detections. This correlation could be explained by a dominant role of far-infrared (FIR) pumping in the H2O excitation. Modelling reveals the FIR radiation fields have warm dust temperature $T_rm{warm}$~45-75 K, H2O column density per unit velocity interval $N_{rm{H_2O}}/Delta V gtrsim 0.3 times 10^{15}$ cm$^{-2}$ km$^{-1}$ s and 100 $mu$m continuum opacity $tau_{100} > 1$ (optically thick), indicating that H2O is likely to trace highly obscured warm dense gas. However, further observations of $Jgeq4$ H2O lines are needed to better constrain the continuum optical depth and other physical conditions of the molecular gas and dust. We have also detected H2O+ emission in three sources. A tight correlation between $L_{rm{H_2O}}$ and $L_{rm{H_2O^+}}$ has been found in galaxies from low to high redshift. The velocity-integrated flux density ratio between H2O+ and H2O suggests that cosmic rays generated by strong star formation are possibly driving the H2O+ formation.
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