SULAYMANIYAH, Iraq—Iraqi Kurds have pushed forward with renewed vigor to retake land from Islamic State in recent weeks, territory that could be used as bargaining chips in future negotiations they are seeking to achieve more autonomy or even independence.

Iraqi territory under Kurdish control has expanded by about 50% in the past two years. Since last month, Kurdish Peshmerga forces took control of a number of villages near Mosul and some strategic access points leading into the city. The offensive came as Iraqi forces and...