If you use a linear regulator, then its got to dissipate (Vin - Vout) X Iout. Your are dropping 11V at 4 amps, this is 44W any linear regulator will dissipate this power as heat. You could put a large wire wound resistor in series with the input to the regulator. The maths goes, leave 2V for the regulator to work with, so Vin = 7V, Vsupply = 16, so the resistor must drop 9V at 4 A = 36 W, but the regulator now has to dissipate (7V -5V) X 4A = 8W. your heat sink could be big enough for 8W.
The sophisticated thing would be to buy a buck regulator. This is a switched mode power supply in a chip, that needs an inductance and a few extra components to get from 16 V to 5 V at 4 A with a power loss of, say 5W or less.
Frank