If the unit Tideturner wants to switch positions with gets bounced by Starcrossed in response, does the Tideturner still move to the other unit's original location?

If Tideturner's triggered ability is on the chain and the friendly unit it is targeting is returned to its owner's hand by Star-Crossed, Tideturner will not move.

Why the movement fails

According to Rule 359.3.e.2, a target is illegal as a spell or ability resolves if it has changed zones to a non-board zone (such as being returned to a hand). When Star-Crossed resolves, the targeted unit leaves the battlefield.

When the ability attempts to resolve, it checks for the location of the targeted unit. Because that unit has moved to a non-board zone, its location is returned as "null." Consequently, the instruction to move the targeted unit cannot be executed, and the instruction to move Tideturner to that location is ignored.

Sequence

1. Tideturner is played (e.g., from hidden), triggering its "When you play me" ability. You choose a friendly unit at another location as the target. The trigger is placed on the chain.
2. Star-Crossed is played in response by the opponent, targeting your friendly unit and an enemy unit. It is placed on the chain above the Tideturner trigger.
3. Resolution (Star-Crossed): Star-Crossed resolves first, returning both your targeted friendly unit and the enemy unit to their owners' hands.
4. Resolution (Tideturner Trigger): When the Tideturner ability attempts to resolve, it checks for its target. Because the unit is no longer on the board, it is an illegal target (Rule 359.3.e.2). The instruction to move the unit fails because its location is null, and the instruction to move Tideturner to the target's location is likewise ignored.

Tideturner remains at its current location, and the targeted unit remains in the hand.

Note: I could not fully verify this ruling against the retrieved rules.