Here's what I have done for years, and rogues getting access to Vex proved it was balanced to me (updated for 2024):
Basic fighting with two weapons: when you make an attack with a light weapon in your main hand, you may make a second attack with a different light weapon with your off-hand. The second attack does not add your ability modifier to damage, unless it is negative. If you have extra attack, you may make an off-hand attack for each main hand attack.
(2d6+Str greatsword balances with 1d6+mod+1d6 for two shortswords. The ability to use two magic weapons and stack bonus damage is the same as stacking a belt of giant strength and a magic two-hander, so I'm not concerned about magic weapons disrupting it.)
Two-Weapon Fighting (Style): You can fight with two weapons as long as both weapons are not two-handed or heavy, they do not have to be light.
(1d6+mod+1d6 upgraded to 1d8+mod+1d8 is +2 damage, and is balanced with Duelist. This has the added benefit of making TWFing Warriors stand out from Rogues and everyone else, because Warriors can use one-handed weapons.)
Nick (Mastery): You can fight with two weapons with a Nick weapon in your off-hand, even if your main hand weapon is not light. (Scimitar loses Nick and gains cleave, but also loses light and goes to 1d8 damage; scimitars aren't small).
(This allows for Rapier+Dagger)
Dual Wielder
General Feat (Prerequisite: Level 4+, Strength or Dexterity 13+)
You gain the following benefits.
Ability Score Increase. Increase your Strength or Dexterity score by 1, to a maximum of 20.
Ambidexterity. When you take the Attack action on your turn and fight with two weapons, you may add your attacking ability modifier to your off-hand weapon damage (usually Str, or Dex if using a finesse weapon).
Quick Draw. You can draw or stow two weapons that lack the Two-Handed property when you would normally be able to draw or stow only one.
(Great Weapon Master let's you add your proficiency bonus to damage with heavy weapons, +2 - +6. This allows you to add your ability modifier to your off-hand damage, likely +3 - +5, but can be pushed to +7 with belt of giant strength or other ability scores boosters. I consider that fair; plus it doesn't give a bonus action attack.)
I still might adjust Dual Wielder, and change the name to Two-Weapon Master. The damage matches up with great weapon master, but I think it should have a defensive component, like being able to forgo offhand attacks for an AC boost, or just a constant +1 AC boost, or getting one free opportunity attack per round since you have two weapons to threaten with.)