Jet aircraft, and the contrails they create, definitely have a more-than-noticeable effect on the weather.
See this story:
What happened to the weather when the skies went empty.
Data indicates that contrails reduce the day/night temperature range.
But the effect is due to water vapor, not CO2. So the results persist only for as long as there is aircraft activity - then dissipate in a matter of a day or two or three.
This was noticeable in the US after the 9/11 shutdown of the North American airspace. And since then, I have taken closer looks at cloud formation and noticed that it is common for high clouds to be dominated by a criss-cross pattern that most likely evolved from contrails.
There is also an issue of ozone.
Nitrogen Oxide emissions from jet aircraft create ozone in the troposphere, but assist in its destruction in the stratosphere. Jets are flying higher to save fuel and are spending more time in the stratosphere.
I don't know of anyone tying this to a measured drop in the ozone layer. I found a 20-year-old article from "New Scientist" that suggested there might be a real problem, but that is not enough to convince me. I would be much more impressed with an article from "Nature", "Scientific American", or perhaps NASA. But articles from those groups do not suggest there is anything to be concerned about.
Finally, there is CO2. Of course, aircraft create CO2 - and in the next decades we could be seeing SSTs which produce more CO2. But this is not a problem specific to aircraft.