Yes, a bound follows from the following theorem.  For a graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint cycles, and let $\tau(G)$ be the minimum size of a set of edges $X$ such that $G-X$ has no cycles.  

**Theorem.** There exists a function $f(k)=O(k \log k)$ such that for every graph $G$,  $\tau(G) \leq f(\nu(G))$.  

This is actually an exercise in *Diestel's* graph theory [textbook][1].  In your context, $\tau(\Gamma)=d$ (take the complement of a spanning tree), so we get $\nu(\Gamma) \geq f^{-1}(d)$.  As noted by  Gjergji Zaimi in a comment to the other answer, this bound is best possible (up to a constant factor) due to a classic example of Erdős and Pósa.  Indeed there is huge body of related work, which are all called "Erdős-Pósa theorems."  See this [survey paper][2] of Raymond and Thilikos or this [webpage][3] for more information.


  [1]: http://diestel-graph-theory.com/
  [2]: https://arxiv.org/abs/1603.04615
  [3]: https://perso.limos.fr/~jfraymon/Erd%C5%91s-P%C3%B3sa/index.html