Dillon Gee (3-5) gave up four runs on seven hits over six innings in his longest and most effective start of the season, but homers by Mitch Moreland and Nomar Mazara helped Texas sweep the Royals 5-3 on Sunday.
The Royals have allowed an AL-high 141 homers this season.
They finish the month of July losing eight of its last nine games and 10 of its last 11 road games. The Royals went 7-19 in July, their worst month since recording the same record in July 2012.