Finding bridges with an added condition in O(V+E)

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Hello,

I have recently been doing this problem 701F - Break Up.

While it does not have an explanation in the official editorial, some user made an informal editorial.

The solution goes like this

I wrote Tarjan's for finding bridges, but then realized such a solution doesn't even pass the samples because a bridge is only guaranteed to split the graph into two components, but not necessarily separating the vertex s from t. How can I modify the algorithm to only consider bridges that split s and t?

UPD: Ok. The solution I tried was to dfs rooted at s, and for each dfs call update where t is reachable from the current vertex. It gets WA on 98. Is the thought process behind this correct? (Or is different solution intended) I want to make sure I'm not looking for nonexistent bug.