This is a layman question.
If heat is the motion of atoms, how can a fast moving water current be cold?
The motion of molecules that is responsible for heat content in water is random motion; that is there are molecules moving in all directions. The directed motion that you are considering ( all molecules moving in the same direction) from the flowing water does possess kinetic energy, but it is not heat energy.
However, if the water flow encountered some obstacle that impeded the flow and randomized the motion of the molecules, some heating would occur. In that case some of the kinetic energy of the flow would be converted to heat energy ( internal energy) of the water.
It may be worth noting though that the average magnitude of the random velocities of the individual water molecules is quite high (probably on the order of 500-600 m/sec for room temp water ) compared to the likely velocity of any "fast moving water current", so the heat energy or internal energy would be large compared to the kinetic energy of the water current as well. Thus, not much heating would be expected from converting the flow to randomized motion.
The speeds of a water current aren't high enough to make any real difference in kinetic energy.
The reason for water currents are that different water temperatures have different densities and so a temperature difference drives the motion of the water. So major water currents often flow between regions of different temperatures. Eg the Alaskan current brings cold water down the Californian coast.
Small water currents, like a hose pipe, feel cold compared to standing water on your skin for the same reason an air flow does. You skin warms up the stationary water immediately in contact with it - then as the water approaches the temperature of your skin the flow of heat reduces and so there is less cooling effect and you don't feel cold. With flowing water you are constantly warming a new bit of cold water and this constant heat flow from your skin makes you feel cold