We develop a theory for manipulating the effective band structure of interacting helical edge states realized on the boundary of two-dimensional time-reversal symmetric topological insulators. For a sufficiently strong interaction, an interacting edge band gap develops, spontaneously breaking time-reversal symmetry on the edge. The resulting spin texture, as well as the energy of the time-reversal breaking gaps, can be tuned by an external moire potential (i.e., a superlattice potential). Remarkably, we establish that by tuning the strength and period of the potential, the interacting gaps can be fully suppressed and interacting Dirac points reemerge. In addition, nearly flat bands can be created by the moire potential with a sufficiently long period. Our theory provides an unprecedented way to enhance the coherence length of interacting helical edges by suppressing the interacting gap. The implications of this finding for ongoing experiments on helical edge states is discussed.