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."
Guntunge has a point. Hungarian and English have different word orders (or conceptual orders), which are not considered in this course. The result is often Hunglish, not English.
Whereas Hungarian moves from the large to the small, from the general to the specific; English moves in the opposite direction. While the Hungarian sentence reads "We go to the river to the turtles," the normal English sentence would be "We go to the turtles by/at the river."
Perhaps we are going to "see" the turtles, as vvsey says, but only the context can tell us this. We might be going to talk to the turtles, to photograph them, to read them bedtime stories, to ask their forgiveness, or to eat them. Without the context we can't know the purpose of going to them. We only know that we are going to them.
Hunglish word order is also a problem in the many nem X, hanem Y structures seen in this course. Hungarian might say "A ház nem piros, hanem kék," but English would never say "The house is not red, but blue." Natural English requires "The house is blue not red" or "The house isn't red; it's blue," depending on the emphasis.