The behavior of water in the deep supercooled regime has attracted significant interest, motivated by the hypothesis of the second critical point of water. Previous studies indicated the existence of a water anomaly, characterized by a minimum in the thermal conductivity of water. Here, we employ nonequilibrium molecular dynamics computer simulation and the TIP4P/2005 water force field to investigate the thermal conductivity of supercooled water targeting four different isobars, 1, 200, 700, and 1200 bar. We demonstrate using NEMD simulations the existence of minima in thermal conductivity associated with the maximum isothermal compressibility and the minimum speed of sound in water, hence establishing a firm connection with the second critical point of liquid water. Moreover, we demonstrate that thermal gradients polarize supercooled water with a thermal polarization coefficient of several mV/K. We explain the thermal polarization effect using a theoretical formulation introduced recently that connects the thermal polarization effect to the isobaric thermal expansion coefficient.