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What is intersecting the river network?
What ever you do, using the off the shelf geo-processing tools will result in a new a FeatureClass of split polylines.
I am guesing that the problem is that your 'network' is not properly intersected.
that is: at a river branch, one line continues on past the intersction and the other line ends there.
You want both of the branches to end at the upper end of the joined stream (as well they should).
You could, I supppose, perform the featute to line (creating a new feature class),
delete all the affected features from the source feature class,
and append the output features back into the source....