The paper

Harant, Jochen; Rautenbach, Dieter; Recht, Peter; Regen, Friedrich. [Packing edge-disjoint cycles in graphs and the cyclomatic number][1]. Discrete Math. 310 (2010), no. 9, 1456--1462

constructs graphs for which the difference between $d$ and the nmaximum number of edge-disjoint cycles is $k$ for any $k \in \mathbb{N}$. If I understand the question correctly this shows you can't have a bound desired without some restrictions on the graphs (I have not thought more about the graphs described in the comments).

  [1]: https://doi.org/10.1016/j.disc.2009.07.017