  The conventional wisdom suggests that transports of conserved quantities in
non-integrable quantum many-body systems at high temperatures are diffusive.
However, we discover a counterexample of this paradigm by uncovering anomalous
hydrodynamics in a spin-1/2 XXZ chain with power-law couplings. This model,
classified as non-integrable due to its Wigner-Dyson level-spacing statistics
in the random matrix theory, exhibits a surprising
superdiffusive-ballistic-superdiffusive transport transition by varying the
power-law exponent of couplings for a fixed anisotropy. Our findings are
verified by multiple observables, including the spin-spin autocorrelator,
mean-square displacement, and spin conductivity. Interestingly, we further
quantify the degree of quantum chaos using the Kullback-Leibler divergence
between the entanglement entropy distributions of the model's eigenstates and a
random state. Remarkably, an observed local maximum in the divergence near the
transition boundary suggests a link between anomalous hydrodynamics and a
suppression of quantum chaos. This work offers another deep understanding of
emergent anomalous transport phenomena in a wider range of non-integrable
quantum many-body systems
