There's no specific measurement given

There are a few descriptions of the bridge, but none provide a measurement. However, it stretches across the River Rhoyne and

had no rivals save for the Bridge of Dream in the Rhoynar festival city of Chroyane.
^{-The World of Ice and Fire, Volantis. George R. R. Martin}

It is so epic in fact that

the Long Bridge of Volantis stands today as the longest bridge in all the known world. Lomas Longstrider named it one of the nine wonders made by man in his book of that title.
^{-The World of Ice and Fire, Volantis. George R. R. Martin}

Due to the difficulty and the tides of the Rhoyne, the bridge took 40 years to complete.

Tyrion describes it as such:

The gateway to the Long Bridge was a black stone arch carved with sphinxes, manticores, dragons, and creatures stranger still. Beyond the arch stretched the great span that the Valyrians had built at the height of their glory, its fused stone roadway supported by massive piers. The road was just wide enough for two carts to pass abreast, so whenever a wagon headed west passed one going east, both had to slow to a crawl.
^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 27 (Tyrion VII). George R. R. Martin}

So I'd say it's fairly long, but not too wide; given that the Long Bridge crosses at the southern delta of the River Rhoyne of which it is said:

Mother Rhoyne waxes so wide that a man upon a boat in the center of the stream cannot see a shore to either side.
^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 14 (Tyrion IV). George R. R. Martin}

1500 metres (on TV) - And yes, I know you wanted a book answer, sue me.

The bridge depicted in the show is approximately 1500+ metres long. We see the size of various human-scale objects such as doorways and people in the close-up shot and from that we can extrapolate that each arch (from the centre of each pillar to the centre of the next pillar) is 37 metres.

The longer shot shows us that there are 13 arches (with possibly one more occluded by some trees in the foreground) which gives us a total figure.

The real location in Spain is only 247 metres long, which gives you an idea of the 'epic scale' that the makers were hoping to inject into the filming through the use of CGI.

---

To keep this answer on-topic (re: books), please enjoy this picture from The Lands of Ice and Fire by George R.R. Martin. If you can work out the scale on this, you're a better man than I.

Speculative guesstimate #1: 3 Km

Book information

@DavidW notes the following sequence of events in DoD Tyrion VII:

1. Jorah and Tyrian ride into Volantis; it is dusk.
2. Jorah and Tyrian ride South within the city, at some point slowing "to a crawl".
3. Tyrion gets chains fitted.
4. Jorah and Tyrion walk to the Long Bridge (but this is not necessarily the same or similar distance to what they covered on the way South)
5. Jorah and Tyrion cross the Long Bridge (i.e. walk its length).
6. Jorah and Tyrion take a room at some house, eat, and go to sleep.

After all that, they have a large part of the night to sleep.

Real-world information

Preferred Human walking speed is about 5 Km/h (Wikipedia). Let's assume Tyrion walks at... what? 25% of that, being a Dwarf and shackled? Let's be conservative and say 20%.

Speculative assumptions

• "The night" is 11 hours overall (to be on the conservative side, and without knowledge of how seasons work on Planetos in terms of day lengths).

• It took J&T 3 hours overall from the city gates until coming onto the Long Bridge. This has essentially no basis, except the assumption that a blacksmith would not want to work well into the night for people he doesn't know.

• Let's interpret "a large part" as less than half, since otherwise it would probably not have been mentioned that they didn't get a reasonable night's sleep. So, no more than 5 hours sleep for J&T at the end of the night.

Calculations

Time spent walking the Long Bridge in hours: 11 - 3 - 5 = 3 hours.

Distance covered: ( 20% * 5 Km/h ) * 3 h = 3 Km.

This is already quite impressive (and double the length suggested in @Valorum's answer based on the TV series). A less conservative estimate might put the length at somewhat longer than that.

Speculative guesstimate #2: 5 Km (or 2.5 Km)

Information from the books

There's a quote of Lomas Longstrider, describing the bridge (see here; probably from the World of Ice and Fire), saying that the bridge was said to support "A Thousand Elephants".

Real-world information

Elephant facts tells us Elephants are 10 feet - 22 feet in length, i.e. 3.3 to 6.8 meters or so (Asian Elephants are much longer than African ones). Can't find width information, but judging from this, the width is about half the length, give or take something for the ears.

Speculation

If we interpret the "Thousand elephants" quote to mean you could literally place 1,000 elephants on the bridge, that gives you an indication of its length. Problem is, we don't know whether the Elephants are supposed to be two-abreast or single-file. Let's assume one-abreast, because it stands to reason that a length estimation would put them in a single file; plus, I doubt you had elephants walking in proper formation, even with the splendor of Volantis.

Let's take an elephant-to-elelphant length of 5m, all included (nose, tail, clearance).

Calculation

1000 elephants * 5 m/elephant = 5 Km

If we assume two-abreast, this becomes 2.5 Km.

Approximately 7.44 km (~4.6 miles)

If we take as fact that:

Mother Rhoyne waxes so wide that a man upon a boat in the center of the stream cannot see a shore to either side.
^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 14 (Tyrion IV). George R. R. Martin}

We can do some math...

From Wikipedia, the equation to determine distance to the horizon is D = √(2Rh), as long as we're dealing with relatively low altitude observations and ignoring things like atmospheric refraction. D is the distance to the horizon, R is the radius of the planet, and h is the observation height.

From Atlas of Ice and Fire, indicates Planetos is a little bit bigger than the Earth at 6920 km (~4,300 miles).

We'll work in meters, because a man sitting in a boat could quite easily have his eyeline at a height of 1 meter and that makes the math even easier. D = √(2Rh) becomes D = √(2*6,920,000) → D = 3,720, 3.72 km. Double that because our boat man is in the middle and we get 7.44 km (~4.6 miles).

Obviously, if the bridge isn't quite at the mouth or the planet isn't actually that big, or the quote isn't accurate, or the boat man is sitting higher, the precise value will change, but the equation won't.