Tungsten is actually denser than lead, even denser than depleted uranium as a matter of fact. It's just super hard, harder than steel. That's why tungsten shot is actually a matrix of some soft material that contains small particles of tungsten to give it weight.
Comparison of densities of various metals in grams per cubic centimeter:
Osmium 22.6
Iridium 22.4
Platinum 21.45
Tungsten 19.35
Gold 19.32
Uranium 18.95
Lead 11.35
Silver 10.5
Bismuth 9.75
Copper 8.96
Iron 7.87
Soft, extremely dense, and non-toxic, gold would make the ideal shot if only an ounce of it didn't cost more than most shotguns.