In my book series, there are 2 main continents on the planet Aurea: Koumaris and Louzi. There is a spot here where the 2 continents come very close to touching, separated only by about 2,500 feet (760 metres) of water at the strait's narrowest point. However, the strait sits on a transform fault and is just over 375 feet (115 metres) deep here. The civilization on the planet, is based primarily on Komnenian-era Byzantium, and as such they would have the architectural technology of around that place and time. Could they bridge this body of water, and if so how?
The thing here is that a bridge isn't required to have supports all 375 feet deep if it doesn't need supports. Or, that is to say, so long as it can just float on the surface, then the bridge could work. And now I introduce pontoon bridges. Pontoon bridges can, given the right equipment, be built very quickly and over wide stretches of water.
Unfortunately, there are two downsides. The first is that it can't be used to transport large quantities of heavy objects, given that it's a floating bridge. And the second is that, given the fragile nature of it, honestly all it takes are a few medium sized waves and the whole bridge is gone. So if this gap is anything but quiet waters, you can have a bridge last a few hours or something, and then it's gone.
But it's possible to build a pontoon bridge.
Even with today's technology building a bridge over a transform fault** is a perilous endeavor - non-trivial displacements over time preclude a rigid bridge and the presence of the fault will trigger major earthquakes frequently. At the best today one may try a suspension bridge, but it won't last long if it ever get to be build at all.
As for a suspension bridge during Byzantium times, some problems of the cables:
rope won't do, not tensile enough, tensile steel is well in the future
any cable/rope used is going to be heavy at those distances - tensioning them correctly is a huge problem at that time (don't tension the correctly and you have Tacoma Narrows)
corrosion - a google search for Golden Gate Bridge corrosion see for yourself
They could, but they wouldn't.
From technical point of view there is nothing that could stop them from building such structure. Justinian build 430 metres long bridge around 500 AD.
There are few reasons why not from logic, logistic and need point of view.
Logic - is there a need to "waste" time, effort and materials to build such structure. Are there no other more imporant need on both sides (like fortification, roads, coast line).
Logistic - Is the movement of people and goods so intense it require a bridge (that would need to be high enough to let ships pass or have bascule part (which again require even more materials). So if the X amount of uses are so big that it cannot be fulfilled by ferry with Y capacity taking of each set of time? So if the ferries would need to take of so often they would form a floating bridge by itself.
Need - bridges over rivers where very strategic points, they created natural bottlenecks. The downside that it was created on both ends. Do you need to spend a lot of money and work on service just for someone on the other side to say "nah, you can't pass. untill you pay us one million dollors!"
What could be done, much cheaper and faster, is to expand existing harbor into to sea. Just 100 metres from each side shorten the distance by 200 metres. Giving you on both sides 200m metres of side ferries, ships and other can attach to. And more space to load, unload good and people.
If by bridge you mean "giant pile of rocks", then yes you could build a bridge, though it would take many decades. A trapezoid with a height and width of 115 meters, a length of 760 meters and an upper width (walking path) of 10 meters would have a volume of 5.46 million cubic meters. The great pyramid had a volume of 2.5 million cubic meters, even more if you include its 2 neighbors, so it is within the realm of possibility that a civilization could spend 50 years dropping slabs of stone into the ocean to very slowly create a bridge.
Absolutely. Technically, it's no problem. You just make the strait less deep. You start dumping fill (largish rocks, mostly - no soil) at one end, and when the level of the fill gets to within about 10 feet of the surface you start building a "normal" bridge, which gradually gets extended as the fill ridge extends. Assuming a 45 degree slope for the fill ridge, you're talking about 10 million cubic meters of rock, which is "only" about 4 times the volume of the Great Pyramid of Giza.
It won't last, of course. Transform faults aren't stable, so the bridge is going have problems - but that's an issue for later generations to deal with. It is not a surprise that most of the 7 wonders of the ancient world are no longer with us.
In your case, this would required extensively large amount of raw materials and hours of works, but it does not seem impossible if your empires are able to commit on this task for a very long period of times.
I once enjoyed a very decent fish dinner on a restaurant on the Galata bridge in Istanbul, a floating bridge which spans two continents as described. Pontoon bridges have been around since at least Roman days.
Sadly, I see it was damaged by fire in 1992 and replaced by a far more boring structure.
I'm sure that they could figure out that they could make a series of traveling floating bridges that could depart from one side of the straight and travel to the other using some form of locomotion. Making a large piece of cloth to catch the wind might do, or they could fashion some sort of stick that they could put in the water to push the floating bridge in the right direction.
If only we had a good name for such an amazing traveling bridge...