PRobably the biggest threat to bighorn sheep in the west is domestic sheep and the many diseases they carry that fell bighorn.
When Euro-Americans colonized the west, the widespread and large bighorn population almost died out, not because of hunting, but because of disease. Over the years bighorn have struggled back, but recovery seems to have stalled out and disease is again one of the main factors. See: "Bighorn sheep pneumonia lingers. Wyoming Game and Fish swabs animals' tonsils to test for deadly disease." By Rebecca Huntington. Jackson Hole News and Guide.
Given that fact that one bighorn sheep tag is can fetch $46,000 (this is not unusual), one would think bighorn would get some kind of priority over domestic livestock, especially on public lands, but this is hardly the case.
Due to the immense political bias in favor of the sheep industry, domestic sheep are allowed to spread disease and push bighorn off the range. This is currently happening in Wyoming south of Jackson where the U.S. Forest Service is actually reintroducing domestic sheep to a beleaguered bighorn range. See: "Sheep grazing proposed for Bridger-Teton National Forest" Associated Press