GAZIANTEP, Turkey — The stream of U.S. weapons heading to moderate rebel groups in Syria is being offset by a fresh torrent of cash for Islamist extremists, much of it from small networks of Arab donors who see the Syrian conflict as a step toward a broader Islamist uprising across the region, U.S. and Middle Eastern officials say. The private donors, who use Twitter and other social media to collect millions of dollars from sympathetic Muslims, are providing crucial backing for Islamist militias that appear to be gaining ground in northern and eastern Syria, even as fighting stalls elsewhere, the...