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With the help of a functional renormalization group, we study the dynamical breakdown of scale invariance in quantum Weyl gravity by starting from the UV fixed point that we assume to be Gaussian. To this end, we resort to two classes of Bach-flat backgrounds, namely maximally symmetric spacetimes and Ricci-flat backgrounds in the improved one-loop scheme. We show that apart from a genuine IR fixed point that is reached at a zero value of the running scale, the renormalization group flow also exhibits bouncing behavior. We demonstrate that the IR fixed point found is IR-stable in the space of the considered couplings. As a next step, we analyze physics in the broken phase. In particular, we show that in the low-energy sector of the broken phase, the theory looks like Starobinsky $f(R)$ gravity with a gravi-cosmological constant that has a negative sign in comparison to the usual matter-induced cosmological constant. We discuss implications for cosmic inflation and highlight a non-trivial relation between Starobinsky's parameter and the gravi-cosmological constant. Salient issues, including the scheme independence of the IR fixed point and the role of trace anomaly, are also discussed.