that's quite a complex question. A lot depends on the flow velocity as well as fluctuations in river level reflecting flow rate or run-off. Generally speaking, if you have a very wide river with low flow velocity and only small temporal variations in flow rate such as across the seasonal cycle as well as a low incidence of extreme discharge/run-off events, the answer would be yes. But as soon as one of these variables changes, the whole picture of river bank erosion and its dynamics changes, too!
PS: Please also consider variables such as river structure. Large - or a big number of smaller - river bends, e.g., will also affect erosion activity. In sum, river bank erosion rates are not just a function of river width, but depend on a complex interplay of a number of different potamological and geomorphological parameters.
I do not think so. Maybe on a given river, but this is also hard to tell. Many variables interplay in bank erosion as was said earlier, one important is river bank geometry (bank height, slope angle), but also bank material. In a study in England we found that clay banks could be very high and stable with low erosion rates while gravel banks were eroding up to to 2m/year in certain places. But of course you need to consider a wide range of other factors, including changes in flow regime, shear stress etc.
I think the simple answer is no. For example, consider the Rio Negro where it joins the Amazon near Manaus. Two nearly equally wide, one is silt laden while the is not. In this case one originates in the easily eroded Andies which is much different from other branches from the Guyana Highlands or the Central Brazilian Plateau. So width alone is not particularly relevant.
From a stream restoration point of view, shallow banks tend to erode less than steep banks. In order to lesson slope, the streams are widened to minimize erosion. This is the opposite of your query.
As has been suggested by prior responses, the interplay of a variety of biological, geological, and hydrological factors--all acting over different temporal and spatial scales--influence erosion rates in stream systems. A small handful are discussed below. Full disclosure: I am not a hydrologist or a fluvial geomorphologist. Take what I say with some skepticism using your local experience and research this topic for yourself, as it applies to your particular river systems!
One important factor--the connectivity of river flows to a surrounding floodplain impact the erosional characteristics of the conveyance channel flowing through it. If a wide floodplain exists, channels tend to be braided or have a main channel with a fairly low width to depth ratio. A system can evolve over geologic timescales to reach an equilibrium state, with points of disequilibrium scattered through the network as random processes occur and cause such changes.
Another factor--vegetation and surficial geology influence the resistance that channels and floodplain surfaces can offer to such flows. Watershed sediment inputs can influence channel networks, with excessive amounts carried by large flood events causing wholesale resetting of channel networks, a phenomenon that will undoubtedly become more common in some places (and less common in others) as hydrologic regimes across the planet are reset due to global warming over the next 100 to 200 years.
Finally, don't underestimate biology! Biological communities are often considered a green veneer of dependent variables that are controlled by abiotic factors in watersheds. This can be a mistaken assumption as there are keystone species--North American Beaver is one where I work--that have enormous impacts on hydrologic and geomorphic conditions in the basins where they live. Such organisms (or their absence!) might be influencing the conditions you're observing now.
Finally, remember that the stream conditions you're observing now are representative of a time slice in the long geological history of a particular river. What you see now may not be stable or ecologically desirable, even if it's been around in your landscapes for decades. Keep an open mind about what should be present in a particular landscape. It might be radically different from the conditions you're observing now. Good Luck!
I really wish it was that simple but in my understanding riverbank erosion is a live process and keeps on happening over the years. It could be because of various reason slope, velocity of water, stream type, kind of restoration (if done). Process of aggradation and degradation is constant sometime it even makes the river and stream to changes its path.
As discussed in the above responses, you have to mention many conditions and consider different parameters for answering this question. For example, it is very important to know if this question is related to a bend or straight route of the river?
But as a quick response, we expect less erosion on banks in the wider river if every condition is equal in these two cases. Considering constant discharge in both cases, widening the river reduces the velocity. Therefore, it decreases erosions on banks. Be careful! Erosion on the walls is different from on the bed. In this scenario, erosion will increase on the bed by decreasing depth.