High-temperature superconductivity is often found in the vicinity of antiferromagnetism. This is also true in LaFeAsO1-xFx (x <= 0.2) and many other iron-based superconductors, which leads to proposals that superconductivity is mediated by fluctuations associated with the nearby magnetism. Here we report the discovery of a new superconductivity dome without low-energy magnetic fluctuations in LaFeAsO1-xFx with 0.25 <= x <= 0.75, where the maximal critical temperature T-c at x(opt) = 0.5-0.55 is even higher than that at x <= 0.2. By nuclear magnetic resonance and transmission electron microscopy, we show that a C4 rotation symmetry-breaking structural transition takes place for x > 0.5 above T-c. Our results point to a new paradigm of high temperature superconductivity.