Here's some math. All damage-per-round calculations done assuming 70% to hit, except for cases where a feat is chosen instead of an ASI (65%), or when using GWM (40%). MH=Main Hand, Off-H = Off-Hand, 2H = Two-Handing.
Level 2 fighter options:
-TWF Fighting Style with two shortswords ------ 1d6+3 MH, 1d6+3 Off-H, AC 16 (Chain Mail) - DPR 70% hit = 9.1
-Dueling Fighting Style with longsword+shield:- 1d8+5 MH, AC 18 - DPR = 6.7
-Defense Fighting Style with longsword+shield: 1d8+3 MH, AC 19 - DPR = 5.3
-Defense Fighting Style with greatsword: ------- 2d6+3 2H, AC 17 - DPR = 7
-Dueling Fighting Style with greatsword: ------- 2d6+5 2H, AC 16 - DPR = 8.4
In future analysis I'll be assuming all non-dual wielders take the Dueling FS for simplicity
Dual wielders do more DPR at the cost of AC. Relatively balanced. Roughly 1 less AC -> 1.1 more damage per round.
HOWEVER, this requires them to use their BA. So other fighters can shove/potion/etc for free while Dual Wielders will sacrifice 4.5 DPR to do so.
Level 4 fighter options:
-Dual Wielder Feat with two longswords: ---- 1d8+3 MH, 1d8+3 Off-H, AC 17 - DPR = 9.75 (only 65% to hit)
-Sword and Board who increased Str+2: ------ 1d8+6 MH, AC 18 - DPR = 7.4
-Greatsworder who increased Str+2: ---------- 2d6+6 MH, AC 16 - DPR = 9.1
-Greatsworder who took and uses GWM: ----- 2d6+14 MH at a -5 to-hit, AC 16 - DPR = 8.4 (plus occasional extra damage from BA attack; let's say 20% of the time you can make an extra attack -> extra 1.4 DPR)
The Dual Wielder is comparable to the Sword&Boarder, but has more AC than the GWM fighter, and has both more AC and DPR compared to the non GWM-greatsworder. Dual Wielder > Greatsworder.
However, again, dual wielders must sacrifice their off-h attack (losing 4.9 DPR) to use a different BA, where other fighters get a free BA most turns.
Level 5 fighters with Extra Attack:
-Dual Wielder Feat with two longswords: ---- 1d8+3 MH x2, 1d8+3 Off-H, AC 17 - DPR = 14.6 (only 65% to hit)
-Sword and Board who increased Str+2: ------ 1d8+6 MH x2, AC 18 - DPR = 14.7
-Greatsworder who increased Str+2: ---------- 2d6+6 MH x2, AC 16 - DPR = 18.2
-Greatsworder who took and uses GWM: ----- 2d6+14 MH x2 at a -5 to-hit, AC 16 - DPR = 16.8 (extra damage 30% of the time because with 2 attacks you're more likely to kill an enemy -> extra 2.1 DPR)
In every single case, dual wielders do less DPR. Either they do ~similar DPR but have 1 less AC, or in exchange for 1 more AC than greatsword wielders, they do ~4-5 less DPR (instead of the ~1 DPR per AC in previous levels).
So now Dual Wielders do less total damage AND must sacrifice even more DPR to use a different bonus action - drink a potion, shove, hide, etc.
This ^ trend only gets worse at level 11 when fighters get their third attack.
*This analysis does not include magic items, doing this with a rogue (sneak attack changes things significantly) or other classes, dipping, hunter's mark, the effects of frequent Advantage or Disadvantage, and possibly more.
