BEIJING—After years of meteoric growth in China, the U.S. company that owns Pizza Hut and KFC is starting to stumble in its single biggest market as intense competition and fast-food fatigue cause Chinese consumers to lose their appetite for fried chicken and pizza.

Yum Brands Inc.'s share price sank 9.9% Friday after Chief Executive David Novak warned that same-store sales in China would drop 4% in the fourth quarter, compared with a jump of 21% the same period a year earlier. Yum's stock is still up nearly 14% year-to-date....