3 attacks (4th is capstone and i'm assuming we DON'T mean max level necessarily) with a holy avenger greatsword against an ogre will deal about 76 damage a turn assuming no expanded crit range and a +13 to hit (due to +3 weapon). 2 attacks will deal about 51. an ogre has 59 hp. at level 20 (so 4 attacks) but without a holy avenger (so no 2d10 radiant - but still a +3 weapon, because i am a merciful god), that's 58 damage - just barely enough to not kill the ogre.
so, uh, unless i'm missing something, then no. the bolded is false. they can typically kill one ogre in one round...if they have a holy avenger greatsword.
that might have been me - and my initial calculations assumed beefy magic items (namely the same holy avenger greatsword as this post!), and i said without those magic items things would be even worse.
You're not including action surge or feats like GWM. Also depends on what "high level" means. Take a level 15 fighter, 20 strength, +1 greatsword. They're +11 to hit, doing around 31 points of damage per round ignoring all the other things I should be adding based on subclass. I'll probably still hit 80% of the time even with a -5 to get +10 damage so it only takes 2 hits because the ogre has 59 HP. I have 3 attacks and an action surge, so 6 attacks for round 1. Yep, I likely take out 3 ogres in 1 round. Perhaps I could do one more if I used some battle master maneuvers or had pole arm master and used a halberd instead of a greatsword for that bonus action attack.