AEW will be on TNT and it was a fine deal for them to make. One of the reasons that they were able to get such a good deal was because there was more than one deal on the table.
Dave Meltzer noted on Wrestling Observer Radio that AEW is having their production paid for by TNT which is an amazing part of the contract. They might not be getting a rights fee, but not having to worry about production costs is an outstanding thing for a new company.
AEW negotiated their deal very well because they were also talking to Showtime. However, it was said that Showtime might not have wanted a weekly television show from AEW. The premium cable channel might have wanted two hour specials every couple of weeks instead of a weekly show.
Obviously, AEW picked TNT because they have much more exposure, but the fact that they also had a backup plan in Showtime dramatically helped out their deal.