MARJAH, Afghanistan—Ten days into the fight for Marjah, U.S. and Afghan troops continue to seize ground, often battling the Taliban from one mud-walled compound to the next. But progress has been slower in winning over local civilians, many of whom are unsure which side will make life safer for their families.

The Marjah offensive—the biggest since the Taliban regime fell in 2001—is being conducted on fronts both military and social.

It's a high-stakes operation. Kabul's international backers have ready tens of millions of dollars in aid for Marjah, and the Afghan authorities have promised to make the town of 75,000—which has been under Taliban rule for years—a model of good government once the fighting stops.

In essence, Marjah is the test case for U.S. President Barack Obama's argument that more troops and smarter counterinsurgency tactics can salvage the Afghan war.

The town measures roughly six miles by 12 miles, most of it thinly inhabited compounds and farm fields, sprouting with alfalfa, opium poppy and other crops.