Fighters (and only fighters) should get a bonus to damage equal to their proficiency bonus gained in Fighter levels. So, a 9th level Fighter gets +4 to each damage roll but a Cleric 4/Fighter 5 would get +3 to each damage roll.  When coupled with extra attacks, this would go a long way toward making higher level fighters' damage scale properly with their level.

My guy, they're the only class in the game that gets 4 extra attacks, and they get twice as many ASI/Feats as other classes.  We don't need to reinvent the wheel.  Just give them powerful weapons that deal great damage and have cool abilities and bobs your uncle.

You're basically just making them barbarians by giving them that ability, because that's exactly what rage does.

Fighters are quite effective in combat at high levels as is, unless you're comparing them to spellcasters who are being allowed to blow their whole load in only 1-2 encounters a day. In that case the solution is longer days.

I can happily rant for paragraphs about the caster-martial imbalance, but that issue is about utility/out-of-combat effectiveness, not raw combat damage.

Yeah, while fighters do have issues, those issues are not fixed by Moar Damage; fighters already do a ton of damage.

I think giving Fighters an extra "social/Non-combat FEAT" would better serve the class, rather than more damage.

Hell, I'd even go so far as to give them MARTIAL ADEPT as a free starter feat, so they can start off being able to possibly do more than swing-hit-damage. I know that's a Fighting Style option, but . . . 

Fighter is a pretty strong class in particular their high number of attacks works really well with a magic weapon. Trust me that + 3 sword is plenty for a high level fighter. 

What the martials really lack is that out of combat problem solving casters get and I don't know if they will ever really be able to compete.  The nature of spells is basically you get many abilities that share a resource. You'd have to give fighters many out of combat abilities to match that. 

Proficiency bonus wouldn’t work since it scales by character level. A 1-level fighter dip gets it, and consider how many people like a 2-level fighter dip for action surge, giving them more damage, too, makes it even more ridiculous. You probably want something based on class level. 
But as others have said, the bigger issue with fighters is they need more out of combat options, and possibly more tactical choices in combat. Damage isn’t the problem. 

I wouldn't recklessly change class abilities because it seems good at the time if I were you. I've made that mistake in my games, and it never turned out well.

If it ain't broke, don't fix it.