BRUSSELS — On the street, Malika El Aroud is anonymous in an Islamic black veil covering all but her eyes. In her living room, Ms. El Aroud, a 48-year-old Belgian, wears the ordinary look of middle age: a plain black T-shirt and pants and curly brown hair. The only adornment is a pair of powder-blue slippers monogrammed in gold with the letters SEXY. But it is on the Internet where Ms. El Aroud has distinguished herself. Writing in French under the name “Oum Obeyda,” she has transformed herself into one of the most prominent Internet jihadists in Europe. She calls...