Bridge Example

Consider this graph: $1-2-3-4$ where edges are $1-2$, $2-3$, $3-4$. Every edge here is a bridge. Remove any edge, and the graph splits into two pieces. This is a path graph with no cycles. Now add edge $1-4$, creating a cycle $1-2-3-4-1$.

Now no edges are bridges because every edge has an alternate path around the cycle. Bridges only exist outside cycles. Any edge in a cycle has a backup path if it fails, so removing it does not disconnect the graph.