Interesting bit of trivia: if you drive through Dayton, Ohio on Interstate 75, you will cross over the Great Miami River five times over a driving distance of less than five miles.
Widths - offhand, I’d give a score in feet of bridge per mile of river (or meter per kilometer). That would take into account the width of river spanned by the bridges, which may be more important for this than the actual average river width. Clearly, the measure you are proposing is going to be very different from greatest number of bridges, or greatest sum of bridge lengths for an entire river. Of course, then we have to agree how we measure bridge lengths.
If you wish to take into account the amount of traffic that crosses a bridge, we could ask about the most CROSSED river in the world, in people per day, I suppose.
Chicago has the most movable bridges of any city in the world. The cite also says there are about 180 bridges in the county. (cite)
There is this advertisement for United Airlines which has a neat display of a bunch of the bridges all going up at once along a particular stretch of river in Chicago.
It’s fun to watch the sailboats move in and out from the lake at the beginning and end of summer. They don’t make the bridges go up like in the commercial but will raise one after another in succession to let the sailboats get to and from winter dry docks.
I strongly doubt it’s the Nile. Once you get south of Cairo, I’d expect crossings to get rather sparse, rather quickly.
Heh. Wiki does have several lists of bridge crossings. The “long small river” argument probably has merit:
The upper Mississippi has more than I care to count: List of crossings of the Upper Mississippi River - Wikipedia
The Lower Mississippi has 20, removing the 10 ferry crossings listed: List of crossings of the Lower Mississippi River - Wikipedia
The upper river includes LOTS of bridges over the Mississippi where it’s a dinky little creek after emerging from Lake Itasca: Google Maps
The Ohio:
If we consider that the actual main stem by volume of the Mississippi River drainage is the Mississippi - Ohio - Allegheny, we have to consider the Allegheny, which also has a large number of bridges because it starts as a small creek:
And here’s the list of crossings of the Danube - it’s pretty large, and I’m not going to count them either: List of crossings of the Danube - Wikipedia
Somebody can poke around these wiki lists, decide which ones are “real” bridges, and count, if they like.
And a boat.
While bridge density is interesting, I was thinking total bridges - hadn’t really thought of tunnels.
But I figured the winning river would go through many cities so it would have sevearl high brideg density spots
Brian
but then the question would shift to “which city has the most bridges?”
Without crossing any twice? 
Kant! 
I happen to live in the city where the first bridge was built over the Yangtze – in 1968! China’s come a long, long way since then, but I’m too lazy too back up my assertion that they probably haven’t beat multiple countries’ spans across more historically-relevant rivers.
This page lists 62 for the Yangtze (65 counting tunnels) with another 8 under construction.
The first bridge listed is at Yibin. Don’t know if that is because there are no bridges above that or if that is just a cutoff point used in the article for some reason.
Whoa. There are 27 in Chongqing alone!
At the end of the list are pictures. Does it not look like they looked at a bridge catalogue and said, “We’ll have one of everything”?
I counted something like 140 bridges across the Danube. Note that I lost count several times, so that’s just an estimate.
The Wikipedia page for the Amazon River claims that there are no bridges over the Amazon!
Indeed I checked Google maps for large cities, and there’s nothing. While it drains mostly rain forest, I would still suggest that the Amazon has to be the most un-exploited large river in the world.
OK, I counted the Mississippi page (see my post above). Similar estimate gives 150 for the upper, with 20 for the lower - about 170. At this point, I would tentatively give the Mississippi the prize. Even the ruse with lower Mississippi - Ohio - Allegheny combined doesn’t seem to beat it, nor the Missouri.