If someone wanted to build such a thing, it would almost certainly be a pontoon bridge composed of floating sections joined together. The average depth of the Atlantic is about 3300 m so I doubt it would be feasible or cost-effective to drive supports into the ocean floor. (OTOH an artifical island on the mid-Atlantic ridge might be a useful anchor point.)
The other advantage of a pontoon bridge is that its sections could be decoupled during severe storms -- effectively they would become ships which could ride out the storm, with passengers and vehicles on board, and then be joined back together.
The world's longest pontoon bridge is the Evergreen Point bridge, the floating section of which is 2310 m long. You are proposing a bridge approximately 1000 times longer.
Evergreen Point cost at least \$127 million at today's prices. So simplistically, an Atlantic bridge would cost \$127 billion. In practice it would be much more, because of the need to make it sturdy enough to withstand the weather in the open sea and decouple if necessary. If we multiply the cost by 10 to be safe, that is $1.27 trillion. There might also be difficulties with sourcing enough metal and other materials to build the bridge.
As it happens, \$100 billion is the approximate cost of the International Space Station. For the higher figure, the estimated cost of the 2003-14 Iraq war was $2 trillion. So given a large enough effort, the construction could probably be financed.
Would it be cost effective? Almost certainly not, unless something happened to radically change our economy and/or technology. For comparison, the world's container ship fleet numbers more than 6800 vessels, averaging 500 m in length. Simplistically, you could just string together almost all of the world's commercial shipping to build your bridge. But it's really a lot more useful to sail the ships around separately, to load and unload cargo at different ports around the world, instead of having a single (probably overcrowded) span from Canada to Spain.
As mentioned in the original question, a Transatlantic Tunnel, either floating beneath the surface or resting on the seabed, is another possibility. It would avoid problems with weather, and possibly be easier to find materials for, but it would still be a massive engineering task and astronomically expensive.