The previously-linked chart appears to show the rate-of-change of the airfare, by itself. It shows how airfare prices themselves went up or down over time, as a percentage, without taking anything else into consideration. That can be useful data, but it is not at all the kind of inflation-adjusted chart that we're used to seeing.

Your link shows charts that include airfare prices integrated with inflation. That's closer to the kind of charts that we're used to seeing when inflation comes up. But it doesn't say which Consumer Price Index (there are many!) was used for adjustment, so that's annoying; it's hard to establish what it means exactly. It's also lacking in granularity and history.

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I usually find that FRED is better for this stuff. The generated charts include citations for source data and also whatever maths were used in the final presentation.

Here's a chart from FRED of airfare price index over time, adjusted by inflation using the usual All Items, Urban Consumers price index. (There's some error included here since that latter index includes transportation and transportation includes airfare, but it seems like that's in the realm of 1% or so and I can only work with the tools that I have.)

https://fred.stlouisfed.org/graph/?g=1XRwq

The vertical scale set so that 100 = June 1, 2026 because that's what I picked (but anyone can pick a different date for 100 using the Edit Graph link; the shape of the line stays the same).

Anyway, it shows that flights are pretty cheap ~right now. The data doesn't show that it's cheaper-than-ever (that happened in May of 2020), but it's pretty darned cheap anyway even with the ongoing war in Ukraine and the stuff that has been going on in Iran.

Are these prices comparing equivalent services? Shrinkflation is very much a thing in the airline industry as well.