d3, d4, d5, d6, d7, d8, d10, d10+1, d10+2, d10+3
Damage is only a part of the equation. Accuracy (percent hit) and durability (opponent hit points) matter, too.
Obviously, increasing the damage for the Fighter (and not other characters) means they're doing more damage, but it is not clear to what effect. Getting an extra d6 per round (3.5 damage on a hit) doesn't necessarily change how long it takes to drop a 10 hp creature or a 20 hp creature, depending on the rest of the damage.
Ie, a Fighter doing 1d10+10 is going to drop a 10-hp opponent with one hit without another +1d6 damage. The odds of dropping a 20 hp foe in a single hit increase from 10% (rolling a 10 on damage) to 45% (rolling a combination of d10+d6 that totals 10 or better or 27/60). But against an opponent that had 30 hit points, the extra d6 wouldn't matter unless the fighter had poor damage rolls.
A lot of game design is like that - someone thinks giving someone a benefit is going to solve some perceived problem, but once you dig in, it might not be the solution that you expected it to be. It's my take that a lot of people try to fix something like the fighters by giving them a damage bonus, but the that becomes the excuse for why they can't provide something else. Ie, if we 'already fixed the fighter', then you don't need to do another fix. But a little extra damage probably doesn't solve the problem.
Take a fighter (any edition) and a troll (any edition) and you'll see that adding more damage is probably not going to give the fighter anything resembling a fighting chance. Unfortunately, D&D (and clones) don't typically give fighters tactical options like impaling a troll on a boar spear and holding them at bay. Ultimately, there's a disconnect between the fiction and the game mechanics - fighters fight with intelligent using terrain, tools and tactics against monsters who are stronger, tougher and have bigger weapons. You can make a Fighter who has an AC, hit points, and damage equal to a dragon, but most people don't REALLY think that's the right level of balance.