I have been resisting contributing towards this thread until now.
The problem is not with steady-state wind, as has already been pointed out by John Farley in an excellent post.
The problem is that there is no such thing as a steady wind.
If you are flying into a gusty wind, your groundspeed will not vary directly with the wind gusts because of your momentum. So turning downwind will affect your lift.
This is, basically, the same phenomenon that can trick people flying into a microburst. As you get near, the headwind increases massively and rapidly, but your groundspeed does not change much immediately because you have inertia providing momentum to be lost first. So people see their ASI getting all excited and cut power. Just at that moment, as their airspeed is adjusting itself naturally, the engines reduce thrust, overall energy decreases - and then the headwind drops, reducing airspeed further, and the downdraft hits them. Low engine power, low airspeed, low momentum and massive downdraft and the ground stops their fall.
You cannot consider instantaneous airspeed to be a simple function of grounspeed and windspeed as there are fluctuations due to gusts and the (sometimes) cushioning effect of your momentum. So when you turn downwind, a gust of wind up the chuff will rapidly reduce your airspeed.
The downwind turn phenomenon does exist - no matter how much common sense dictates that it should not.