Count Bridges

First, we need to create a new graph $G$ removing all the bridge(s)/cut-edge(s). Now, we need to construct a tree $T$ from the new graph. For each connected component in the graph $G$, create a node for the tree $T$. And for each node from the original graph, keep track of which node from the original graph, belongs to which node of the tree $T$.
We can complete the above operations in $O(n log(n))$ time complexity.
For two nodes of each bridge of the original graph, find which node of the tree $T$ belongs to and connect those nodes. The new graph $T$ we've created in this way will be always a tree. Now, create a sparse table using this tree $T$.
For each query, find the distance between $A$ and $B$ using their LCA. The distance is the answer.
We can create the sparse table within $O(n log (n))$ time complexity and answer queries in $O(log(n))$ time complexity.