Pure-quartic solitons have gained significant attention recently due to their ability to achieve higher energy than classical solitons for short pulse durations, leveraging the interaction between self-phase modulation and anomalous fourth-order dispersion. However, challenges in generating the required dispersion profile and the scarcity of sufficiently low-loss devices with high nonlinearity and negligible nonlinear loss have restricted experimental progress. In this paper, we report a class of pure-quartic Bragg solitons that balances self-phase modulation and the ultra-strong Bragg-grating-induced negative fourth-order dispersion in combination with negligible group velocity dispersion and negligible third-order dispersion. We demonstrate pure-quartic Bragg soliton-effect compression of 2.4× in a compact millimeter-scale integrated low-loss and highly nonlinear waveguide circuit. Our findings show the potential of exploiting the exceptional dispersion profile of nonlinear Bragg gratings for advanced soliton generation and pulse shaping, particularly the advantageous energy scaling and associated compression of pure-quartic solitons.