For any regularity exponent $$\beta <\frac{1}{2}$$ , we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $$C^0_t (H^{\beta } \cap L^{\frac{1}{(1-2\beta )}})$$ . By interpolation, such solutions belong to $$C^0_t B^{s}_{3,\infty }$$ for s approaching $$\frac{1}{3}$$ as $$\beta $$ approaches $$\frac{1}{2}$$ . Hence this result provides a new proof of the flexible side of the $$L^3$$ -based Onsager conjecture. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $$L^2$$ -based regularity index exceeding $$\frac{1}{3}$$ . Thus our result does not imply, and is not implied by, the work of Isett (Ann Math 188(3):871, 2018), who gave a proof of the Hölder-based Onsager conjecture. Our proof builds on the authors’ previous joint work with Buckmaster et al. (Intermittent convex integration for the 3D Euler equations: (AMS-217), Princeton University Press, 2023.), in which an intermittent convex integration scheme is developed 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.