The bucket feels heavier out of the water because it is no longer supported by the buoyant force, and you feel its full weight.
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You feel lighter because of the buoyant force exerted by the water, which partially supports your weight.
Pressure exerted in any confined mass of fluid is transmitted undiminished in all directions.
Fluids exert pressure on the base and walls of the container in which they are enclosed due to their weight.
A solid exerts pressure on a surface due to its weight.