To rival the performance of modern integrated circuits, single-molecule devices must be designed to exhibit extremely nonlinear current-voltage (I-V) characteristics(1-4). A common approach is to design molecular backbones where destructive quantum interference (QI) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) produces a nonlinear energy-dependent tunnelling probability near the electrode Fermi energy (E-F)(5-8). However, tuning such systems is not straightforward, as aligning the frontier orbitals to E-F is hard to control(9). Here, we instead create a molecular system where constructive QI between the HOMO and LUMO is suppressed and destructive QI between the HOMO and strongly coupled occupied orbitals of opposite phase is enhanced. We use a series of fluorene oligomers containing a central benzothiadiazole(10) unit to demonstrate that this strategy can be used to create highly nonlinear single-molecule circuits. Notably, we are able to reproducibly modulate the conductance of a 6-nm molecule by a factor of more than 10(4). The conductance of a six-nanometre molecular wire can be reproducibly modulated by a factor of more than 1 x 10(4) at room temperature by enhancing destructive quantum interference amongst occupied molecular orbitals.