If enemy Elder Dragon targets a unit at each battlefield and then I react with Tideturner to swap my units, does Elder Dragon still kill the targeted unit I moved?

No — your units live.

Tideturner's trigger resolves first (LIFO), so by the time Elder Dragon's play effect resolves, each originally targeted unit is at a different location than the one it was chosen at. Elder Dragon's ability is a series of per-location restrictions ("deal 1 to up to one enemy unit at battlefield A, ... at battlefield B, ..."), and those choices are locked in at finalization. A unit that left its chosen location no longer fulfills that restriction, so the instruction affecting it is ignored — it takes no damage (Rules 359.3.e.5, 359.3.e.9; see FAQ #10312, same principle as Flash vs. Elder Dragon in FAQ #9919). Its lethal-damage passive never comes into play because the damage is never dealt.