Hi Afsi,
There are a couple ways to move the text along the river.
Using River placement you have the multiple curves but you can't control where along the line the label places along the feature. The label engine will choose the most appropriate spot depending on the curves. In this case you can convert the labels to annotation and then use the follow feature mode in editing to drag the annotation feature along the curve to where you want it.
If you change to the Regular placement with a curved offset position - you won't have the multiple curves you'll just have one curve, but you'll have the ability to set where along the line the label should be placed.