I have come to my senses. I was thinking that all cost modificatition was always in effect. So Bridge would lower WF's cost to $5, but WF would set its cost to the last gained card, let's say $3 (from a gained Smithy now affected by Bridge), but then Bridge would lower it to $2. But if WF's own ability is also still in effect (just like the Bridge's), then it would reset itself to $3. Etc.
So yes, they are in conflict, and one has to win.
EDIT: So this means that (contrary to what I was saying earlier) we do need a special rule here. From the cards we can't know whether Bridge or WF takes precedence. But the rule could of course also had been that cost reductions take precedence, meaning that WF would end up costing $2. I feel like this would be a bit easier for players to grasp, and what people intuitively feel would happen: just always apply cost reducers.