A couple of points to consider -
Only Jaime is wearing armor.
Most humans are naturally buoyant. SCUBA divers have to wear weights to allow them to sink, so while Jaime clearly sinks from the weight and density of his armor, it's not the same as just dropping a rock. He can move and wouldn't necessarily just be pinned to the bottom of the river.
Bronn is clearly assisting Jaime.
We see them emerge down-river.
If Jaime is dragged to the bottom, initially, why do we assume that they fought their way to the surface and swam from there? Wouldn't it be easier, assuming they could hold their breath for a couple minutes, so head directly to shore moving along the bottom of the riverbed, until they got to shallow enough water to catch gulps of air or finally stand?
Anecdotally - As a former lifeguard, we used to compete in a competition each year where, in one of the events, we had to retrieve a diving brick from the bottom of the deep end, and swim it back to the shallow end of the pool. The other competitors always tried to drag it to the surface, where they could breath, and swim it back. I always just used the weight of the brick to allow me to push off the bottom and essentially leap-frog along the bottom until I got to shallow water. It was much faster than trying to fight and swim at the surface while carrying additional non-buoyant weight. I don't think my competition ever even covered half the distance when I hit the finish line.
If I was Jaime or Bronn and I could see the shoreline within potential reach, I'd gather my legs under me and "dive" horizontally and repeatedly to cover that distance as fast as possible. With Bronn treating Jaime as a "weight" and also leaping at the same time, they cover the distance that much faster.
The added benefit of bottom-level locomotion would be that you would be less likely to be seen swimming and thrashing at the surface for an extended period of time. Remember, when Jaime was last seen he was about to be enveloped in a ball of fire that reduced other humans to ash, literally. From Daenerys's perspective, all she saw was the giant expanding fireball between Drogon and Jaime. That might additionally help them not be noticed if no one considered the idea that there might still be a Jaime (though Tyrion seems to make that assumption, so what do I know?).