For Kurdish fighters, last month’s victory over Islamic State militants in the town of Kobani in northern Syria was only the beginning.

Their ambition is to build on an alliance with moderate rebels in Syria and become the chief force fighting the extremists in the country.

Its commanders say such an alliance could be just the partner that the West has been seeking all along in the battle against the Islamic State group, also known as ISIS or ISIL, which began its rise in 2013 and now holds a third of both Syria and neighboring Iraq. […]