this is the Eulerian path problem
If there are more than 2 points with an odd number of bridges then the problem is impossible, when here are exactly 0 or 2 such points then it is possible, with 2 odd points you start at one and then end up at the other, with 0 you come back where you start.
The solution for the path is simple,
Start at a point with an odd number of bridges and add it to your path and remove it from the set, from there move further by picking any bridge forward, repeat until there are no more bridges at your current point.
Then at a point you passed and there are still bridges to cross split your path there and go over a bridge until you eventually come back to that point. Reconnect the path with the one you just followed.