Several reasons for this the most obvious being a ford. Historically travelers chose routes to reach a destination by the shortest way, perhaps an obvious statement but rivers are one of those geographic features that would alter direction of travel because it forms a barrier. The route then deviates to the easiest place to cross, a ford perhaps.
Worth noting if the way is designated, footpath, bridleway, RUPP or BOAT, because this will indicate the historical use of the way if the designation was correctly made for the Definitive Map. This is a designation that might indicate to the walker the level of water expected at a crossing. The Ordnance Survey usually mark fords, but not always.
An interesting post on another forum, some years ago, a member advocated carrying a pair of rubble sacks, these worn over the boots and tied off above the knees provided a lightweight and quick conversion to waders. I tried this on a ford on an upper part of the River Vyrnwy, about 30m of knee deep water over a rather rocky bed. My initial crossing was successful. I then carried two companions and on my final crossing for the kit, a leak had formed and I got a bootful of water. I think the designated way was a BOAT, sadly no boat.
Lower down the Vyrnwy river there is an anomaly, where a RoW footpath appears to cross the river but there is deep water. Apparently after the 1962/63 winter a huge block of ice in the melt water carried away the old stone bridge.
Another anomaly appears, where rights of way lead to a defunct Ferry. When the ferry ceased, made redundant by a nearby road bridge the Rights of way where not diverted to the new crossing. A particular example being Bridge Solars, crossing the River Wye in Herefordshire.
Another sad anomaly on the Welsh Border with Shropshire is the Maginnis Bridge, here there are rights of way on both sides of the River Severn at Pool Quay but no right of way over the Bridge. This is the only non highway bridge for many miles either way over the river.