For any regularity exponent β<12, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class C0t(Hβ∩L1(1−2β)).  By interpolation, such solutions belong to C0tBs3,∞ for s approaching 13 as β approaches 12.  Hence this result provides a new proof of the flexible side of the Onsager conjecture, which is independent from that of Isett.  Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an L2-based regularity index exceeding 13.  The proof employs an intermittent convex integration scheme for the 3D incompressible Euler equations.  We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.