How to quickly, easily cross a river with the horse?

When the land is criss-crossed by a lot of rivers, travel on a horse becomes a chore. Small rivers can be jumped over... if you time your jump right. All larger ones - dismount, attach lead, swim across, walk some more to get the horse out of water, return, detach lead, pick lead, mount. And twenty seconds later the river turns around and you need to repeat the whole dance.

It's a horrible chore that really detracts from the joy of exploring on horseback. I'm fine with dragging the horse behind a boat across the sea, as it's not nearly as frequent, but I'm sick of the rivers.

Any better way to quickly, easily get a horse across a small span of water?

It appears that, while riding the horse, you cannot make use of the the frost walker enchant (either on your boots or on the horse's armour) as neither will freeze the water. However if you dismount and attach to the lead you can happily walk over the river and not have to worry about your horse falling into the water (as the ice freezes for long enough for it to cross).

The easiest way is to do what people do in real life: look for a fordable part of the river. The width of the river doesn't really matter, if it's only 1 block deep. You can ride a horse through 1-high water without a problem. Even if parts of the river are deeper, as long as the deep sections are jumpable, you are good to cross, because horses can jump while standing in 1-high water, too. This does require taking a few detours to look for the best spots.

The quickest Minecraft way to cross a river on horseback is to build a 1-block wide bridge. This could even be done from horseback, in some cases, although you can't use the backwards sneak method while riding. Even if you dismount to build the bridge, there's less fuss than the dismount-lead-detach lead-remount method.