United States Steel, which said it was restarting idled furnaces around the country in response to the tariffs, has filed objections against a slew of companies seeking exclusions, sometimes opposing dozens of requests from the same source. In most filings, U.S. Steel said it was capable of producing the metal for tubing, casing, piping and sheeting that companies asked for permission to import.

The Commerce Department must still deal with nearly 20,000 petitions in its queue. While the administration has said the exclusions are an effective way to ensure the fairness of the tariffs, companies that have applied for the exclusions criticized the exercise as both long and disorganized.

“This is the most screwed-up process,” said Mark Mullen, president of Griggs Steel, a steel distributor in the Detroit area. “This is a disservice to our industry and the biggest insult to our intelligence that I have ever seen from the government.”

Mr. Mullen has made 90 requests and has not been told the status of any of them. He is waiting to submit 2,000 more requests, “but they take so damn long,” he said.

As the chaotic exclusion process continues, Mr. Trump is expanding his trade efforts on multiple new fronts, threatening tariffs on up to $450 billion of products from China and roughly $350 billion of imported autos and auto parts.

Although the president has famously said trade wars are “easy to win,” the lingering difficulties of the steel and aluminum tariffs, which were put into place three months ago, show the inevitable complexity of trying to reshape the rules of global trade in a matter of months.

The handful of companies that have received exclusions to the tariffs are so far pleased. Bill Brebrick, the United States sales manager at Zapp Precision Wire, which was granted an exclusion for a zinc-coated flat-rolled wire that it produces in Germany, said the outcome had shown him “that the system’s working.”