JOHNS CREEK, Ga.—The special election in Georgia’s Sixth Congressional District is on its way to being the most expensive U.S. representative’s race in history, already exceeding $37 million with several weeks still to go before the runoff vote on June 20.

Two candidates, Democrat Jon Ossoff, who has never held elected office, and Republican Karen Handel, a former Georgia secretary of state, are spending millions in this suburban Atlanta district of about 700,000, flooding airwaves, boxing out commercial advertisers and, in...