A friend of mine was killed in an Islander a few years back, on base with a very strong crosswind (ie, tailwind for him).
All the ingredients were there- long duty day, late back after an ambulance trip, sub-Arctic weather, and- crucially- high ground under his base leg. He was caught in the illusion (high groundspeed, ground close therefore even higher apparent groundspeed, therefore TAS MUST be high, mustn't it......?).
There is no loss of airspeed in the turn to downwind, other than normal induced drag losses- as Checkboard said. I've tried it in a C152 in a 30 knot wind, and I've tried it in a 146 in a 130 knot wind. Same effect both times- no irspeed change, other than that to be expected in calm air as a consequence of the turn.