KYIV, Ukraine—Russian troops stormed toward Ukraine’s capital Saturday, and street fighting broke out as city officials urged residents to take shelter. The country’s president refused an American offer to evacuate, insisting that he would stay. “The fight is here,” he said. As dawn broke in Kyiv, it was not immediately clear how far the soldiers had advanced. Ukrainian officials reported some success in fending off assaults, but fighting persisted near the capital. Skirmishes reported on the edge of the city suggested that small Russian units were probing Ukrainian defenses to clear a path for the main forces. The swift movement of the troops after less than three days of fighting further imperiled a country clinging to independence in the face of a broad Russian assault, which threatened to topple the democratic government and scramble the post-Cold War world order. Ukraine President Volodymyr Zelenskyy offered renewed assurance Saturday that the country’s …