Why does a swimmer cross a swimming pool in the same time as crossing a flowing river? I read that the time taken to cross a swimming pool (still waters) is the same minimum time required to cross a flowing river, provided the swimmer crosses it the river in the minimum time required, and the pool and the river are of the same width.
My thinking is:

*

*In the pool, the whole component of the swimmer's velocity is forward, so time taken would be $\frac{W}{V}$ where $W$ is the pool width and $V$ the velocity.

*When crossing the river, time taken would be $\frac{W}{V_y}$ where $V_y$ is the component of the swimmer's velocity perpendicular to the width of the river, assuming the river flows along the x-direction. Hence time to cross the river would be more.

Why is this wrong?
 A: As gandalf61 has explained, one way to view this problem is the current flow could be considered orthogonal to the intended direction of travel. With such a view, the minimum crossing time solution is to swim directly perpendicular to the current, while allowing the current to sweep you downriver.
But I will make what is sure to be a contentious claim: this isn't actually the fastest way to swim across a river, and in fact a river with current can be crossed in less time than a swimming pool of the same width, given some leeway with the definition of "swimming" and some generosity regarding the feasibility of the idea.
What you say? How is this possible?!
Here's another physics question: say there is a sailing course on open water with a start and a finish, the finish being directly downwind of the start. At the starting gun, a ballon is released to drift freely in the wind. Can a sailboat, powered by nothing but the wind, make it to the finish before the balloon?
The answer is, counterintuitively, yes. The reason is that the sailboat can generate lift, effectively extracting energy from the wind which the balloon can not. The fastest path for the sailboat is not to sail directly downwind, but actually at some angle away from it, and then to turn periodically, taking a zig-zag path towards the finish.
In fact, very cleverly this technique can be used to construct a vehicle which can travel downwind faster than the wind by turning the sails into a propeller. Such a vehicle is demonstrated in this Veritasium video. Compare the direction of the windsock to the streamer on the vehicle and you can see that the vehicle is in fact traveling faster than the wind.

Now how does this apply to swimming across a river? In a comment, Chris H referenced reaction ferries which use a fixed tether to cross a river. Their propulsion comes not from any power source, but rather by accelerating the river current towards the bank using the rudder, and the reaction force propels the ferry.
This got me thinking, is there any way a swimmer could use the river current to generate lift? If so, it's possible crossing a river could be faster than a swimming pool of the same width. I can imagine there could be many possible techniques, and I'll not claim any of them are feasible, only that they are theoretically possible. Demonstrating feasibility will have to be left to a better swimmer than I.
Here's one such technique: consider that a person simply floating in the river, not swimming, is equivalent to a person floating in a swimming pool with a breeze. We know that sailboats could cross a swimming pool with a breeze, and so a swimmer, if allowed to carry a sail, could extract some propulsive force from the apparent wind through a similar mechanism. If this propulsive force can be attained in addition to the force from ordinary swimming, doing both (swimming and sailing) at the same time will result in crossing the river faster than the swimming pool.
If the swimmer is not allowed a sail, I think we can still not rule out the possibility that the swimmer's own body could not be oriented in such a way as to achieve a similar aerodynamic advantage. I'm somewhat skeptical of the feasibility of this approach, since I'd guess the aerodynamic advantages would be more than offset by the disadvantages of a less effective swimming technique, but at the same time I can't think of a theoretical reason it's not possible.
If that's too far-reached, here's another idea: a swimmer can certainly use the same reactive force that propels reaction ferries, but only until they have been accelerated to match speed with the current. A reaction ferry uses a tether to prevent this from occurring, and a swimmer has no such tether, but that does not mean the current is useless, only that it is only temporarily useful. I would think with sufficient practice it should be possible to extract a little bit of lift in the first few seconds after diving into the river, gaining some initial advantage on the swimming pool swimmer. After the dive, the river swimmer would swim perpendicular to the current as before, but that initial advantage would mean the river swimmer would arrive first.
A: Your reasoning would be correct if the swimmer in the river was trying to reach the point on the bank opposite where they started. To do this they have to swim in a direction angled upstream, so relative to the water they have to swim a longer distance than the width of the river.
But to cross the river in the minimum time the swimmer should swim in a direction perpendicular to the banks. The river will carry them some distance downstream, but they will only have to swim the width of the river relative to the water  - which is the reference frame in which their swimming speed is measured. So although they travel further relative to the banks it only takes them the same time as swimming across a swimming pool of the same width.
A: In first approximation, namely when considering the river a medium with constant linear speed, that is a reasonable assumption.  But the flow speed of rivers is not constant: it is higher in the middle than at the banks.  That causes additional drag and Bernoulli forces on a body depending on its shape, orientation, and movement.
It is not uncommon for people to drown in a large river and be found dozens of miles downstream.  A river that would move as a block would be quite easier to exit and would have people wash up at the first slight bend.
So the counterintuitive result from simple physical considerations actually has problems matching reality once we are talking about water that moves at large enough speed.
