In 5th edition there's a problem between using the different weapon types and their effect on the action economy. Most weapon types in this edition, so long as you meet any prerequisites necessary (having proficiency, medium size for heavy, or within the weight requirement if using v. encumbrance), don't cost you anything in terms of the action economy instead of just using the weapons with your action, but two-weapon fighting requires you to use your bonus action, which I feel is an unnecessary requirement that has caused using 2 weapons to fall out of favor for many classes that appreciate the use of a bonus action. This is most apparent with the Ranger, which has been the staple dual-wielding class for many editions, but now hasn't due to it requiring a bonus action that also competes with the spells.

I usually wouldn't consider this a problem and chalk it up to game balance, but I realized, two-weapon fighting (Henceforth referred to as TWF) is one of the weapon types that scales terribly with levels, yet still requires the most out of the action economy. A comparison I like to show is 1 Greatsword vs. 2 Handaxes. For a level one barbarian, the greatsword does the same damage as the 2 handaxes, but the handaxes require you to use your bonus action, which comes into direct competition with Rage (I acknowledge after raging the second handaxe attack and the first will combined deal more damage, but my point is that you are at a deficit). This is just level one though, the more the game scales the more TWF falls behind. At level 5 the greataxe barbarian deals 4d6 a turn (if he hits), and the handaxe barbarian deals 3d6 a turn, but is still investing more into the action economy.

The main point of this post is that while I think that it still should be limited to 1 bonus attack per turn, it shouldn't consume your bonus action, because no other weapon types in the game require that from you, and that's a heavy investment for not much in return. 

My final appeal for why TWF consuming a bonus action should be removed is that from a narrative standpoint, what about attacking twice in a turn is preventing you from raging, & what is preventing you from casting a spell that is only verbal components (as most bonus action spells are) as a Ranger & why wouldn't swinging a greatsword also preventing you from doing the same?

The problem is complicated, and is part of why Mike Mearls hates the bonus action system.  You want to make TWF not be the default, go-to option, and that's difficult.  Compare to a greatsword; out of the box it's 2D6+mod.  At lower levels, when fighters don;t have much to do with their BA, TWF is actually better than a greatsword.  It doesn't scale though.  Additionally, classic weapon pairs such as rapier and dagger are not permitted out of the box because rapiers aren't light, and the light weapon rule /must/ be in effect to keep d8 weapons out of the mainhand without TWF being superior to 2H weapons.

TWF is just a slippery slope, and imo, is why Scimitars and Arming swords were demoted to D6 weapons from their historical D8 perch.

What I would like to see is TWF working kind of like martial arts from a monk.  Your mainhand weapon is what it is, your offhand weapon starts off as a d4, regardless of what it is.  That actually makes sense in the perspective if I hit you with my katana when I swing it one handed with my right hand, it will cause more pain than if I hit you with the same katana swung with my left hand.  As you increase in levels, you get better with your offhand weapon and it begins to do more damage.

For the TWF fighting style, I'd make it swing once, without expending my bonus action.  A cleric, without the TWF style would still have to spend a bonus action to swing the offhand weapon, and the offhand could not swing more than 1 time, unless higher levels require a second swing to keep up with GWF characters.

The tricky part is that two shortswords technically outclasses one greatsword, as while the total damage is theoretically the same (2d6+STR) you're doing it across two attacks so you've got a better chance of doing some damage rather than potentially none at all with one attack.

I wonder if the easier fix would be to allow two-weapon fighting to use either your bonus action or your reaction, i.e- immediately after resolving an attack, you can use either of these (your choice) to make a single extra attack with your off-hand weapon?

This way it's up to you which one you sacrifice for the extra attack, which I think could make sense as you either spend the bonus action to keep the ability to opportunity attack available (or some other reaction ability), or you spend your reaction so you can still use your bonus action, basically overcommitting to a flurry of action in your turn, leaving you no chance to react to anything else.

You are failing to recognize the true benefit of TWF and how powerful it is.   

Among other things,  you are incorrect about Bonus Action = 1 extra attack being uncommon.  That is a standard rule used often.  Monks, Polearm Master, Battle Master's Commander's Strike, Spiritual Weapon Spell etc.  The real point of TWF is not the extra damage one extra attack does, but the extra chance to hit.    Paladins want it for  smites, Rogues want it for Sneak Attacks,  and Champions use it for Improved Critical.  Battle Masters do it with things like trip and push.  Extra attacks = a chance to do something cool, not a simple chance to do a bit more damage.

They did the Bonus Action = 1 and only 1 extra attack for a reason - experience with earlier editions.  Stacking stuff makes for abusive rule possibilities.

You are ignoring the powerful possibilities and concentrating on a sucky, non-maximized idea of multiple hits.  As such you think TWF sucks, when the truth is what you are DOING with it is what sucks.

You do not want TWF, you want something else.  So design it and leave the extremely powerful TWF alone.

I do recognize I mostly pointed out the downsides of TWF and I'll acknowledge its good points, but I'd also like to point out where it falters & why those good points aren't so great. 

The best upside is its ability to synergize with abilities that trigger off of the amount of attacks you make like rage, hunter's mark, hex, & bestow curse. My counterargument for why in its current state isn't the best is that there is negative synergy between most of these abilities and TWF due to both consuming a bonus action, meaning you are going to be going into a damage deficit. Despite from a narrative perspective it making no sense that swinging with your offhand prevents you from casting a spell that only has verbal components, or in the case of a Barbarian, attacking with your offhand means that you can't go into a rage.

I acknowledge as well that with TWF you have a better chance to hit once, but if we are acknowledging that, that also means we have to acknowledge that you are less likely to hit both times with TWF than you are to hit once with a greatsword, meaning yes, you are more likely to deal some damage, but less likely to deal an equivalent amount or still a lesser amount (at higher levels as seen with above damages).
I also should acknowledge that yes, while the bonus action attack does allow somebody to have another opportunity at using their trigger on hit abilities like smites & sneak attack, but it also still detracts from using the core class features if you are using 2 weapons. If you are a paladin you aren't allowed to use one of your smite spells and attack with a second weapon on the same turn, & attacking with a second weapon as a rogue increases your ability to sneak attack, but it also removes all movement versatility that you have with cunning action. Consuming a bonus action is so vital for many classes in the game that if you want to use TWF, you have to give up part of the class's identity to do so.

Another point I should address is the TWF fighting style for Fighter & Ranger, which I fully admit is the best fighting style in the game, but you should also acknowledge that that feature doesn't  synergize with anything, at most what it does is a +5 potential damage per turn, whereas if we look at the dueling fighting style for a fighter, due to it scaling with extra attack, it is a possible +8 damage per turn, possibly +16 with an action surge, and it doesn't rely on your stats at all.

I won't deny TWF is good at lower levels because it has a good start, but as the classes level up, gain extra attack, & gain bonus action abilities, TWF falls heavily out of favor & it turns out incredibly clunky with other class features. I feel making it a once per turn you can make a bonus attack (that doesn't use your ability modifier unless you have the fighting style) without using a bonus action wouldn't break the game, & it makes more sense from a narrative perspective that you can do your class abilities and still attack with your offhand.

Quote from crzyhawk >>

The problem is complicated, and is part of why Mike Mearls hates the bonus action system.  You want to make TWF not be the default, go-to option, and that's difficult.  Compare to a greatsword; out of the box it's 2D6+mod.  At lower levels, when fighters don;t have much to do with their BA, TWF is actually better than a greatsword.  It doesn't scale though.  Additionally, classic weapon pairs such as rapier and dagger are not permitted out of the box because rapiers aren't light, and the light weapon rule /must/ be in effect to keep d8 weapons out of the mainhand without TWF being superior to 2H weapons.

 At our game we permit to off-hand to be a d4 weapon with a d8 weapon in the main hand, so combinations like rapier and dagger or longsword and dagger are allowed without requiring the feat.  I think this is a reasonable adaptation/modification of the rules.  However, the second weapon is only drawn in the second round.