Steel's density of ~7.75 times that of water means you can discount 13% of its weight to buoyancy, but that's about it.
I have lifted 18 kg of collected dropped weightbelts from the bottom on one occasion, which put me at -10 kg of surface buoyancy after my gear weight and wing buoyancy was accounted for. The peak weight was -14 kg at the bottom for the initial ascent, due to wing compression (it was 45 meters down). This was a major physical effort and I had to hand the belts to the boat ASAP, but I was able to stay sufficiently afloat to ask for the assist.
Generally an experienced technical diver wearing jet fins can sustain about 12 kg of upward thrust, with fins. This is useful for an emergency ascent with failed buoyancy devices. Static thrust has been measured at 15-19 kg for ~90 kg body weight professional divers. Producing upward force is not fully equivalent to swimming, as your lateral speed will be low. It's just a struggle to get to the surface, and you could probably brave a narrow stream like that.
This is not the average. The average sustained thrust with fins was measured at about 64-69 N, or about 7 kg. This can be done over a prolonged swim - a river several hundred feet wide.
A practice among good swimmers called "monkey diving" involves wearing no buoyancy compensator and compensating for buoyancy changes with swimming thrust only - so this can be considered a practical swimming weight. The buoyancy at the beginning of a monkey dive is about -4 kg. This takes some effort, but is easily manageable with fins.
Without fins, humans produce far less static thrust. I can stay afloat and swim with the aforementioned -3 kg of buoyancy without fins, but it's exhausting and slows me down. I can carry more very briefly, but -3 kg is as much as I'd be willing to risk carrying across more than 400 meters, being ready to ditch the weight. Over a narrow stream (<50m), maybe -5 kg in a do or die situation. My weight and swimming fitness would be in the range for the character you describe.
Your average medieval soldier was certainly not a skilled diver, or an skilled swimmer, nor did they have any fins at all. This limits their ability to overcome negative buoyancy to -1 kg for most, and maybe -3 to -6 kg for the best swimmers, with a fairly large body for the era. This number is for swims across calm waters; large lakes, very wide or fast rivers, open sea can be challenging as it is (for that reason, everything above and below is for fresh water).
A sleeveless mail vest weighs about 5 kg. Armor is useless without a weapon (another 1-2 kg), so one would bring weapons first, armor second. It's possible to make lighter armor, for instance a steel plate with coverage similar to a SAPI insert. Such small plates were sometimes attached to mail or leather. But it's unlikely that someone would bother creating that just for swimming, since they can just as well make a waxed leather bag that gives one +20 kg of buoyancy.
So the short answer is: Without fins or any buoyancy device - a simple log will do - you can't count on swimming over a decent-sized river with any kind of commonplace medieval steel armor that would be useful on its own.
A competent swimmer without fins can cross rivers with their weapon and their leather armor pieces. With fins, the really good swimmers might be able to also carry a sleeveless mail vest.
A shield would be a good flotation aid, and early styles - low-density lime wood with little metal - can support about the shield's own weight in buoyancy. As such aids interfere with swimming, their buoyancy replaces dynamic swimming thrust rather than add to it, but it's a much more practical way of crossing rivers than rushing it.
If we go outside the military, professional swimmers such as pearl divers did exist at the time, and would be more capable. But finding one that turned soldier would be a one in a thousand occurrence and wouldn't be enough to make an army. Rafts and leather bags are much more practical.
To put an upper bound on what's possible, modern Olympic-level athletes produce about the same static thrust without fins as a skilled diver with fins, so that ability to carry ~10 kg of steel across a river could also be expected of them. This level of swimming fitness takes years of training and only came to exist with the reestablishment of full-time professional athletics in the early 20th century.