Two-Weapon fighting sucks. It shouldn’t. It should be cool. Right Drizzt?

Replace the following rules and feats in the Player's Handbook with the following:

(PHB 195)

Two-Weapon Fighting (Attack Action)

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can also attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the second light weapon, unless that modifier is negative. If either weapon has the thrown property, you can throw that weapon, instead of making a melee attack with it.

Net change: This means an off-hand attack no longer requires utilizing a character’s Bonus Action anymore, and begins to level the playing field on damage output between users of two-weapons and two-handed weapons. Keep in mind this change only allows a single attack per round with an off-hand weapon, not an off-hand attack for every main-hand attack you make. Everybody can do this. And to risk redundancy, this means players are neither required NOR ABLE to spend their bonus action on an off-hand melee attack without other rules in place.

(PHB 72 & 91)

Two-Weapon Fighting (Fighting Style)

You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light, and you may add your ability modifier to the damage of your second weapon. You may draw or stow two weapons any time you would otherwise be able to do so with one.

Net change: This extracts the ‘even if they aren’t light’ portion from the dual wielder feat, making it largely more accessible to characters that desire this playstyle without the feat investiture, and cleans up some rules housekeeping regarding managing a weapon in each hand.

(PHB 165)

Dual Wielder (Feat)

You master fighting with two weapons, gaining the following benefits:

• You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
• Each time you would make an attack, such as with the attack action, an opportunity attack, attacking with the Ready action, or otherwise triggered by an outside source to ‘make an attack’, you may instead attack once with each held weapon. The additional attack feature allows you to attack once with each held weapon per attack.
• If you do not use your action to take the attack action, you may make a single attack with your off hand as a bonus action on your turn (but only a single attack, as a small exception to the above.)
• As part of 'other activity on your turn', you may temporarily hold two one-handed weapons in one hand such to perform tasks or somatic components with your free hand, switch hands with your weapons or return one of two weapons held in one hand to its proper hand. You may not attack with a weapon while it is held in the same hand with another weapon, but you may adjust your grip as such any number of times during your turn as required.

Net change: Allowing an off-hand weapon swing, with EACH main-hand weapon attack instead of just the first, and to swing with both weapons on an opportunity attack, brings two-weapon styled fighters much more in line with the potential damage output of two-handed weapon wielders. It furthermore provides versatility to still get a quick attack in, on turns when you must spend your action doing something else, highlighting the nimbleness of the style, as a counterpoint to fighters swinging big, slow, heavy weapons. Finally, it offers a bit of ability in managing the additional equipment, beyond that in the TWF Fighting Style.)
The sum total of these changes more or less brings 2-weapon fighters into line with 2-handed weapon fighters. It exchanges the few big-hits (esp those enhanced by the +10s of the Great Weapon Fighter feat) for an increased number of medium hits, which… balancing the additional # of attack rolls and chance to miss with the increased times your str or dex modifier are added to damage rolls… and some versatility instead of a cleave. Its not perfect. But its better.