A sixth-level sorcerer has three third-level spell slots and six sorcery points. That sorcerer can get ten more sorcery points by converting their lower-level slots, and then convert ten sorcery points into two more third-level spell slots. (We assume the sorcerer has a few rounds of downtime to do this before the battle.)
The sorcerer then can cast fireball five times, once per round, for an average damage of 28 before saves.
The sorcerer can apply Empowered Spell to each of their casts. This lets them reroll up to four damage dice. The average damage goes from 28 to 33.61 (according to my simulation). Then, using Draconic Fire Bloodline, they can add their +4 CHA mod to the damage roll, for an average damage of 37.61.
A bear has a +0 DEX save, versus DC15, so it passes 30% of the time, which multiplies the damage by 0.85. We multiply by four bears, and we get 127.8 damage per round.
If our race is tiefling, then we can also cast hellish rebuke once for 3d10*.85*.2=2.8 damage per round. The total is 130.6 damage per round.
Can some other build do better?
I think sorcerer is the best class for this, simply because the sorcerer is the only class that can cast five fireballs at this level. Multiclassing would worsen the build because it would lose the Elemental Affinity power which requires six levels of sorcerer.
I did not investigate other sorcerer options — I just took the one that was visible on https://roll20.net/compendium/dnd5e/Sorcerer (Draconic with Fire), as it offered an additional +4 fireball damage and that seemed like a good deal. It may be possible to do better using some other choice.
It’s unfortunate that the sorcerer doesn’t have a good damage-over-time spell to supplement all the fireballs, perhaps making use of their concentration or their bonus action or their reaction. Unfortunately I didn’t see anything promising on their spell list.