Elephant, rhinoceros, &c all have much thicker legs in proportion. The answer, I think, lies in the fact that the animals you mention all evolved as cursorial animals (that is, they run to escape predators). Less mass in the lower leg means it swings easier, so the animal can run faster.
There are two things you're apparently not noticing in that picture. First, the the horse's lower leg is almost entirely bone (and some tendon), and it's bone that does the supporting. The propulsive power comes from the large muscles of the hip, thighs, and shoulders.
Second, the lower part of the leg (with the white wrappings) is not anatomically equivalent to the human's lower leg, but to the bones of the hand and foot. You can see this if you look closely at the rear leg in that picture. The femur, equivalent to the human's thigh, ends at the knee just above the belly line. Then the tibia extends about halfway down, ending at another joint which you might think is the knee, but which is called the 'hock' in horse-speak. The white-wrapped part is a metatarsal, equivalent to human foot bones, then there pastern bones equivalent to human toe bones, ending in the hoof/toenail.
So consider that you can, if reasonably fit, walk around on tiptoe without crushing your foot and toe bones, then imagine the end result of your ancestors having done this for the last several tens of millions of years :-)
PS: With horses, there is some effect from human selection, too. Racing & show breeds tend to have thin lower legs, draft horses & working breeds have proportionately thicker ones. My first horse, a thorobred/arab mix, had legs about as thick as my wrists (granted, I'm a fairly muscular guy); my current mustang, about the same height & weight, has legs about twice as thick.