DAMASCUS—On a weekday in mid-August, traffic clogs the streets of downtown Damascus. Cafes and restaurants are overflowing with customers, relaxing as they smoke narguiles with friends. The souks are packed with shoppers almost every day, and the services industry is booming. In Bab Sharqi, a neighborhood in the old city, at least a dozen bars have opened over the last eight months. Every weekend they are packed with punters, enjoying a few drinks with their food as they listen to the latest Taylor Swift hit, interrupted by the occasional mortar explosion. Less than a kilometer away, the Syrian army and pro-government militias are fighting elements of Islamist and al-Qaeda linked armed groups in a war that will determine the country’s future.