  Starting from an ultraviolet fixed point, we study the infrared behavior of
quantum Weyl gravity in terms of a functional renormalization group (RG) flow
equation. To do so, we employ two classes of Bach-flat backgrounds, namely
maximally symmetric spacetimes and Ricci-flat backgrounds in the improved
one-loop scheme. We show that in the absence of matter fields and with a
topological term included, the effective action exhibits dynamical breaking of
scale symmetry. In particular, it is shown that apart from a genuine IR fixed
point that is reached at a zero-value of the running scale, the RG flow also
exhibits bouncing behavior in the IR regime. We demonstrate that both $\beta_C$
and $\beta_E$ reach the RG turning point (almost) simultaneously at the same
finite energy scale, irrespectively of the chosen background. The IR fixed
point itself is found to be IR-stable in the space of the considered couplings.
Ensuing scaling dimensions of both operators are also computed. Salient issues,
including the connection of the observed bouncing RG flow behavior with
holography and prospective implications in early Universe cosmology, are also
briefly discussed.
