Gene drive technology has the potential to address major biological challenges. Well-studied homing suppression drives have been shown to be highly efficient in Anopheles mosquitoes, but for other organisms, lower rates of drive conversion prevent elimination of the target population. To tackle this issue, we propose a gene drive design that has two targets: a drive homing site where drive conversion takes place, and a distant site where cleavage induces population suppression. We model this design and find that the two-target system allows suppression to occur over a much wider range of drive conversion efficiency. Specifically, the cutting efficiency now determines the suppressive power of the drive, rather than the conversion efficiency as in standard suppression drives. We construct a two-target drive in Drosophila melanogaster and show that both components of the gene drive function successfully. However, cleavage in the embryo from maternal deposition as well as fitness costs in female drive heterozygotes both remain significant challenges for both two-target and standard suppression drives. Overall, our improved gene drive design has the potential to ease problems associated with homing suppression gene drives for many species where drive conversion is less efficient.