The real problem is the fact that the BLM is also responsible for issuing grazing licenses to cattle ranchers on the same public lands set aside for the horses. So the agency can replace every horse they can remove from public land with a fee-paying cow that agribusiness gets to graze, subsidized by the American government. According to two General Accounting Office reports the BLM was making removal decisions not on the actual numbers of horses that a range can support, but on the recommendations of advisory groups "largely composed of livestock permittees".
So the guys who stand to make the most money from having wild horses removed from public land are the ones telling the BLM that horses are the primary cause of overgrazing and degradation of public lands. The truth is that because horses tend to roam and can find forage in areas where cattle and sheep can't, they cause far less harm to a habitat than any livestock.
When cattle graze they don't chew the grass, they pull it from the ground; if the soil happens to be wet they will therefore rip it out by the roots. Horses on the other hand have front teeth, allowing them to crop grass as they graze, so they are far less likely to destroy the root system. A horse's digestive system is actually beneficial to a habitat, because they pass grass seed through their system, thus replanting as they graze.
As to the BLM's claim that horses are degrading grazing lands, horses aren't the critters that defecate in their own water supply; cattle are. Horses aren't the animals who hang out in one area of land until it's stripped clean of any and all forage, necessitating human intervention to move them on to other pastures. Even without any of that information, the numbers don't lie; at current levels livestock out number wild horses by 200 to 1 on public lands. You tell me who is going to have the biggest impact on the environment: two hundred head of cattle standing in one place, or one horse wandering around looking for food?