ISTANBUL—Turkey's dramatically increased economic and political role in the Middle East and North Africa is creating high stakes and awkward conflicts for its government and businesses, as they try to navigate turmoil across the region.

On Monday Turkish leaders were struggling to manage fallout from the latest outbreak of violence, in Libya, as they began sending ferries across the Mediterranean to evacuate some of Turkey's estimated 25,000 citizens living there, most of them construction workers.

Libya...