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Even before the offensive, ISIL cells with headquarters in Mosul showed an ability to unleash a wave of attacks against the Shiite-dominated government by deploying vehicle-borne improvised explosive devices.

“I think a lot of the capabilities they’ve demonstrated show that they are an organized group able to coordinate complex operations in places of their choosing,” Mr. Johnston said. “That’s really the crux and the key to their success more than materiel itself.”

ISIL also has embraced social media like no other terrorist group. Its posts on YouTube are circulated by news media and private intelligence websites, which, in effect, are doing ISIL’s bidding.

“A big part of its bureaucracy internally is a media committee,” Mr. Johnston said. “This is information operations, propaganda type of entity that has been messaging within Iraq since at least 2006.”

An example of the West spreading ISIL propaganda as a source of information is a report from the Combating Terrorism Center at West Point.

“There is already a picture on Twitter of Abu Umar al-Shishani, the military commander of the ISIL in Syria, stepping out of his personal [U.S.] Humvee,” the report notes. “Several posters on jihadist web forums and Twitter have sent out requests for helicopter pilots to potentially fly some of the aircraft that the ISIL captured in recent days.”

The media committee also knows how to meet deadlines. Within hours of its Iraqi conquests, ISIL documented the victories in the English-language Islamic State News magazine. The stories outlined its economic goals for Iraq.

“Virtually all Islamic extremist groups make use of social media to advance their causes, but the ISIL’s media production team is especially adept, and its target audience extends beyond the Arabic-speaking world,” the West Point group said.

The combination of ISIL’s growing power, the sorry state of Iraq’s army and the lack of a Sunni-Shiite governing coalition prompts the highest-ranking U.S. military officer to say this:

“If you are asking me, will the Iraqis, at some point, be able to go back on the offensive to recapture the part of Iraq that they’ve lost, I think that’s a really broad campaign-quality question,” said Army Gen. Martin E. Dempsey, chairman of the Joint Chiefs of Staff. “Probably not by themselves.”

Gen. Dempsey, who directed the training of Iraqi security forces from 2005 to 2007 and recently expressed disappointment in their performance, laid out what now must be done: “You’d like to squeeze them from the south and west. You’d like to squeeze them from the north and you’d like to squeeze them from Baghdad. And that’s a campaign that has to be developed.”