Translation:We go to the river to the turtles.
This "elmegyünk" sounds to me like a longer distance. The river is not across the street or "over on the other side" that we could go "over" to. It is just somewhere, no relation to our current location. So we just go there, plain and simple. So that is why I would not use "over". But I would also omit "away" and just simply say:
"We are going to the river to (see) the turtles."
"To" used twice sounds odd to me, like going to two places.
We go to the turtles at the river.
The turtles are our goal? So "to" should "aim" there and they just happen to be at the river.
I agree with guntunge. I think a better English versions would be we go to the turtles at/by/in/near (which one really doesn't matter, all depends on the exact location of the turtles compared to the river) the river.