Inspired by this question, we can ask a more general question.
Question 1. Let $G$ be a connected graph with $n$ vertices and $m$ edges. Let $C$ be a cycle of $G$ such that after deleting all edges of $C$ the $G- E(C)$ is still connected. Then what is the (best) lower bound of $m$ (on $n$)?
The following question is equivalent to the previous one and may be easier to understand.
Question 2. Let $G$ be a connected graph with $n$ vertices and $m$ edges. If for any cycle $C$, after deleting all edges of $C$, $G-E(C)$ is not connected, then what is the (best) upper bound of $m$ (on $n$)?
Perhaps increasing the connectivity would be more interesting. Maybe we can ask for same question for $2$-connected graphs, even for $k$-connected graphs when $k\ge 2$. More formal:
Question 3. Let $G$ be a $k~(\ge 2)$-connected graph with $n$ vertices and $m$ edges. If for any cycle $C$, after deleting all edges of $C$, $G- E(C)$ is not connected, then what is the (best) upper bound of $m$ (on $n$)?
Question 4. Let $G$ be a $k~(\ge 2)$-connected graph with $n$ vertices and $m$ edges. If for any cycle $C$, after deleting all edges of $C$, $G- E(C)$ is not $k$-connected, then what is the (best) upper bound of $m$ (on $n$)?
$k=2$ is the first step.