“It’s kind of the family looking after each other,” said Mike Seawright, a field coordinator for Doctors Without Borders, whose medical workers treated arriving Syrians for an assortment of ailments like dehydration and diarrhea, but not war wounds.

“To generalize, people here are in pretty good condition,” he said.

The local authorities and aid agencies said they were unprepared for the latest wave of refugees, and that an existing camp nearby in northern Iraq was terribly overcrowded, housing nearly 50,000 refugees in a site built for 22,000. Since last week, nearly 40,000 more Syrian Kurds have come, bringing the total close to 200,000. Officials have scrambled to provide for the latest influx, calling on residents across the region to make donations.

When Massoud Barzani, the president of Iraq’s Kurdish territory, whose security and prosperity has served as a model for the aspirations of Kurds in Syria, Turkey and Iran, visited a new refugee camp on Monday, he was greeted like a revered spiritual leader, his vehicle mobbed as he appeared through a sunroof.

“We are brothers to you,” Mr. Barzani told the crowd. “And you are now in your home and in your country.”

Mr. Barzani has lately sought to position himself as not just the leader of Iraq’s Kurds, but someone who can unite all Kurds as they push for more independence and more democratic rights. He recently threatened to send his own security forces, known as peshmerga, to defend Kurds in Syria. Next month he will host a regional Kurdish conference that is viewed as a setting to discuss how the Kurds can seize the turmoil gripping the Middle East to advance a shared agenda, the distant goal of which remains an independent Kurdish state.

While the flood of refugees has tested Mr. Barzani’s administration, it has also presented him with a political opportunity. By caring for refugees and, in his call to arms for a people long steeped in a martial culture, he is burnishing his credentials among Syrian Kurds as a regional leader. As some refugees crossed the border Wednesday morning they approached a journalist with a video camera and chanted, “Viva Barzani!”

“God bless Barzani,” said Mariam Ahmed, 30, a refugee at the camp near Erbil. “We have everything we need here.”