Yes, a bound of the form $\Omega(\frac{d}{\log d})$ follows from the following theorem.  

**Theorem.** For every integer $k$, every graph $G$ either contains $k$ edge-disjoint cycles, or a set of edges $X$ of size at most $(2+o(1))k \log k$ such that $G-X$ is a forest.  

This is actually an exercise in *Diestel's* graph theory [textbook][1].  Since the minimum number of edges to remove to make a graph acyclic is $d$ (take the complement of a spanning tree), the bound follows.  As noted by  Gjergji Zaimi in a comment to the other answer, this bound is best possible 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 [survery paper][2] of Raymond and Thilikos.  


  [1]: http://diestel-graph-theory.com/
  [2]: https://arxiv.org/abs/1603.04615