Remove bridges to make finding a path impossible.

MathFun123's answer shows that it is possible to remove three edges in this way, but not that three is the minimum.

To see that you can't do it with less than three, consider the following.

There are three completely separate paths from A to B shown. In order to cut off B from A, you need to remove at least one red edge to destroy the red path, and similarly at least one blue edge and at least one green edge. So at least three are needed.

By removing the red lines $A$ and $B$ are isolated; in other words: no path is possible from $A$ to $B$.

"So I considered that we can do this by 'isolating' either A or B"

I tried the same thing at first. As a general approach to problems like this though, you should try to confirm that your supposed methods are actually the most efficient. Isolating either A or B is a good starting point, however, as the other answers have shown you, it's not a complete solution.

"So what is wrong with this method?"

The reason your method failed is you made a faulty assumption: that any two paths must converge only on island B. Were that true, then isolating either A or B (whichever had fewer brides) would work; if the number of paths could only increase until converging on B, then A would have to have the fewest bridges (and vice versa).

However, we know that this isn't true, and in this example, it becomes relevant with the top two paths stemming from island A; they converge on pretty much the next island. This shows us that you can remove one fewer bridge by waiting until they converge on that island than by isolating A immediately.

"How can be the answer, 3, be reached?"

A more effective approach would be to still start where you started: by isolating A. That's at least 4 bridges you have to remove. From that point though, you have to travel each path and see if at any island the number of paths to B decreases or increases. That will help you pick the most efficient bridge to remove for any given path.

For example, lets follow the bottom-most path from A. After it's first island, there is still only one path to B. After it's second island though, this splits into two paths to B. Now that's two bridges you have to deal with instead of one, so we can safely remove the first bridge on this path. The same applies to the second bottom-most path.

As for the upper two paths, we can see the second island for the upper-most path is the same as the first island for the second upper-most path (I really wish they labeled each island). Rather than splitting into two paths, at this island two paths merge into one. This lets us remove both of the upper-most paths by removing one bridge, which takes our total bridges to remove down to 3.

In this example, 3 is the best we can get. However, we can also apply this approach to any graph and achieve the best outcome (it should be noted that just because a path splits into two, it shouldn't be immediately thrown out. There's still a chance both of those paths will converge on another one, that just didn't happen in this example).