Did God turn His Back on Jesus?

I think Jesus certainly felt abandoned, as would any of us in similar circumstances. I do not personally believe that it is possible for God to turn His back on God, so I don't think that happened.

At the end of the day, it does not matter to us what God's precise method was. What matters is that it was effective.

We find things like this difficult in part because it is difficult for our finite brains to comprehend how the Lord Jesus could be a human person in every regard and yet be the God of the universe at the same time. Harder still is to understand how God in flesh could be in any way separate from God in Spirit. The doctrine of the Trinity is the best way we have found to represent this three-yet-one, one-yet-three God, but like all such doctrine it is imperfect due to our limitations.

When Jesus subjected Himself to the will of the Father, He did so as a human person. As a human person He bore the sin of all humanity across all of time and served the death sentence that sin required. As God, (yet still human) the Lord Jesus rose from the dead by the power of God.

Humankind was at enmity with God (separated from the Source of Life).
Humankind was sentenced to death (because separation from the Source of Life IS death).

Jesus was fully human.
Jesus was perfect.
Jesus could therefore serve the death sentence required on behalf of all humans.
Jesus shed His blood and died.

But Jesus is also God.
God is the Source of eternal Life. “For as the Father has life in himself, so he has granted the Son also to have life in himself” (John 5:26, ESV).

Therefore, Jesus could not be held by death.
Jesus rose from the dead for all of us, just as He died for all of us.

He was our representative.
He acted on our behalf.