BRUSSELS — Turkey surprised European Union leaders on Monday by hitting them with a new set of demands if it is to help stem the flow of refugees from Syria and Iraq and other migrants seeking to enter Europe.

Leaders assessed the demands at an emergency summit meeting in Brussels, where Turkey’s prime minister asked for billions of euros in new assistance, easier access to visas for Turks to go to Europe and the dramatic acceleration of talks on Turkey’s membership in the bloc, a discussion that has languished for years.

The toughening of the Turkish position underscored Ankara’s apparent attempt to win more support from Europe if it is going to be expected to protect the bloc from hundreds of thousands of new asylum seekers.