Why can't pure water conduct electricity since it can be reduced at cathode and oxidised at anode?
Indeed, water can conduct electricity by electrolysis. But this is hampered by the fact that water has very high resistance. From Electrolysis of water:


  Electrolysis of pure water requires excess energy in the form of
  overpotential to overcome various activation barriers. Without the
  excess energy the electrolysis of pure water occurs very slowly or not
  at all. This is in part due to the limited self-ionization of water.
  Pure water has an electrical conductivity about one millionth that of
  seawater.


And from Self-ionization of water we can see how small it is:


  Water molecules dissociate into equal amounts of $\ce{H3O+}$ and $\ce{OH-}$, so their concentrations
  are equal to 1.00×10−7 mol∙dm-3 at 25 °C