No. Pure water does not conduct electricity. As a matter of fact, pure water is a good insulator. The waters we encounter in real life are very likely to be impure. This is because they contain a certain amount of dissolved particles and substances.
Most of these impurities are charged and will therefore permit the flow of electricity through the liquid. Pure or distilled water on the other hand is an excellent insulator because it contains no impurities and as such no ions. Only neutral molecules are present and as such, the water becomes unable to conduct electricity.