It all depends on the type of fish and the type of turtle.
I have a large pond behind my home and a combination of both fish and turtles exist together. I do notice when I feed them bread crumbs that the fish will back off when the turtles go for the same bite. I have never seen the turtles advance toward them in any way, but it seems the fish definitely have a respect for the turtles. We did have a large snapping turtle in the pond (different than those I mentioned above) that would eat the babies of our wild ducks that would raise their young. One of my neighbors witnessed the turtle pulling the last young duck under and say mother duck going after the little duck. The large turtle attacked and wounded the mother duck so bad that she did a few hours later. Back to your question, I am not familiar with koi but turtles and fish can co-exist.
No the turtle will kill it or tont it all day I feeed fish to my turtle