Four of the largest U.S. steel producers have made a majority of the objections to companies’ requests for relief from President Donald Trump’s steel tariffs, sinking many importers’ hopes of escaping the duties.

Nucor Corp., U.S. Steel Corp., AK Steel and TimkenSteel accounted for approximately 10,191 objections out of 19,213 submitted to the Commerce Department, according to a Bloomberg Law data analysis. The objections appear to be working: U.S. Steel and Nucor, respectively, prevailed in getting Commerce...