Multi-energy flow state estimation is a critical area of research in the energy internet. Current research on the state estimation of electricity-water coupling system is still in its infancy, facing challenges such as poor adaptability to water network conditions and insufficient utilization of electricity-water coupling information. To address these issues, this paper proposes a bilinear weighted least square (WLS) state estimation method for water networks, considering the correction of the friction coefficient. By incorporating virtual measurements of water pumps and bi-directional transmission of coupling information, a bilinear WLS state estimation method is developed, suitable for interdependent, cooperative and joint operations of electricity-water coupling systems. The effectiveness of the proposed method is verified by an 11-node water network and two electricity-water coupling systems, formed by coupling with IEEE-14 node and IEEE-118 node power systems. Numerical results highlight the necessity of correcting the friction coefficient in water network state estimation, the computational efficiency and adaptability of the bilinear WLS method for low flow rate water networks, and the the improvement in accuracy, data consistency and observability achieved through cooperative and joint estimation for both power systems and water networks.