Rivers help transport sediment from higher regions into the oceans (where it's hard to get back into the atmosphere).
And this sediment could be carbon-rich.
Rivers generally include dissolved oxygen. Therefore any organic content in suspension will tend to dissolve. The movement of water (open to the atmosphere, etc) will tend to entrain some oxygen from the atmosphere.
For carbon to be preserved in sediment, it has to be buried before it can break down. Typically this would occur where water movement is minimal so that it can precipitate out and be buried by mud/sand.
Areas of water may be anoxic (eg. ocean bottom waters, or the sea bottom waters that led to the deposition of the English Kimmeridge Clay), but these areas tend to have restricted currents so that what oxygen that is available is "quickly used up" oxidising some of the organic material.
Yes you can capture carbon in sediments - that is how all fossil fuels are formed, but you have to have the right conditions. Lakes, swamps, seas are much better than running rivers themselves. A deltaic swamp with lots of organic stuff growing in situ but experiencing periodic river floods could be a good way to preserve organic material... (cf. the Carboniferous in Europe and the US).
Interesting research on the influence of iron's oxidation states on the oxidation of organic matter: Preservation of organic matter in sediments promoted by iron, Lalonde et al, Nature v483, pp198–200