Considering the 20th level wizard has an AC that's probably 5 points lower, has probably around 60 HP less than the fighter and cannot bonus-action-heal (second wind), the wizard isn't very high level.
Easily dealt with with just a few spells.
Mage armor gives AC of 13+Dex, meaning you'd need a negative modifier to be so poorly-off, and it lasts all day, so it's one 1st-level spell, a pittance for a even a 10th-level Wizard. They're likely to be much closer than 5 points, though. The best AC you can get without magic items is 21, and that's only if they hyper-specialize in defense at the expense of other things (sword-and-board + plate armor + Defense fighting style). A Wizard with
mage armor and a mere +3 Dex mod--easily achieved by level 12, since Wizards only really value Int, Dex, and Con as stats--is exactly 5 points less than the best possible AC a Fighter can have without magic. And then for emergencies,
shield can handily deal with other problems, which completely covers that gap for the price of another 1st-level spell. Plus, because
shield is only for attacks that actually
hit you and lasts the rest of the round, it's almost never wasted!
The HP gap would be more of a problem if the Wizard weren't so much better at playing keep-away than the Fighter could ever dream of being.
Misty step, invisibility, and (especially)
mirror image all dramatically mitigate that...and those are all 2nd level spells, which while not
as much of a pittance as 1st level ones, are a pretty easy cost to swallow. And then
fly and
blink at 3rd level are even better, and
vampiric touch can even make up for the HP problem in a pinch (though I wouldn't rely on it myself).
Throw
mage armor, shield, misty step, mirror image, fly, and
blink on a 13th-level Wizard. This isn't even half the spells you know of any of those categories, since you gain two spells at every Wizard level after 1st (and six at 1st level!) If you
can learn some spells from scrolls or books, it would of course be useful, but far from necessary. Oh, and even with Arcane Recovery giving you an extra 5th level slot, you still have 2 extra spell levels' worth of refresh, so you can regain a 2nd level slot or two 1st level slots, for a total of four 2nd or six 1st level spells a day. This is not a Batman Wizard who can summon every spell imaginable from a quantum spellbook; this isn't even a particularly comprehensive approach, just a useful one--and you still have 12 other spells you can prepare each day (level 13 + Int mod of 5 = 18 prepared spells each day).
This approach even works great for several Wizard schools. Abjuration, bladesinging, chronurgy, conjuration, divination, and war magic are all excellent. Eocation, illusion, necromancy, and transmutation are all decent, albeit needing a bit more effort to make maximum use of. I think the ideal is probably Diviner or Abjurer, though Conjurer has the fun of teleporting a ridiculous amount of times, and Bladesinger/Warmage are best if actually intending to get into fisticuffs, since they mitigate the AC problem quite nicely. (Indeed, a Bladesinger can actually rival the AC-hyperfocused Fighter while in Bladesong;
mage armor + Dex mod 3 + Int mod 5 = 13 + 3 + 5 = 21; with higher Dex, you can actually
exceed the best non-magical AC a Fighter can achieve.)
So...yeah. You can do a hell of a lot to mitigate these issues, while still having 75% or more of your kit open for utility, power, shenanigans, or just goofing off.
That's why I say a 13th-level Wizard is on a par with a 20th level Fighter. Are there things the Wizard might struggle to deal with still? Sure, we can construct situations like that. But
overall, in general, the Wizard is in a better position, and with even a little bit of foreknowledge and a DM that isn't completely denying any opportunity to scribe more spells, they can be in an excellent position for nearly all situations. The Fighter, even at level 20, has
absolutely nothing they can do to bridge that gap.
Wizards-in-the-generic can put a portion of their resources to being like Fighters, while still having the vast majority of their Wizard-ly stuff and
not needing to take a "specifically becoming more like a Fighter" subclass (though, of course, they can still choose to). Fighters-in-the-generic? Not a chance. It takes
actually becoming part-Wizard (EK), and even then, it's slim pickings.