And I have a i7-3770k at 4.6GHz with gtx970. In skyrim I get 180fps, but if I add mods (I've got 170 loaded) that shoots down to 60. That's all cpu. I can get 300fps in CSGO. Detail settings don't matter, I get that at low or ultra on either game. I'm hitting cpu limits, the gpu is capable of more.
So test settings. Play the game as is. Note the fps. Change to ultra, then low settings. Note the fps. If the fps is changing greatly, that means the fps is limited by the gpu, it's changing according to details/post processing. If fps doesn't really change much at all, then the gpu isn't struggling, the cpu is.
For example, my skyrim. With 170 mods, it's cpu bound. The most I can get is 60fps. I could stick a RTX2080ti in there and still be locked at 60fps. It wouldn't change anything. If the cpu was capable of 200fps, and gpu settings only allowed 60, then yes, a move to a stronger gpu will have benefits, but if cpu is maxed, gpu doesn't help.
Having a 2600 is no guarantee of anything. You can still get middling fps output, and moving to a stronger gpu not show any real change. It's going to depend on your pc setup, ram speeds and location, bios, gpu, game code, resolution, GeForce Experience settings, Xbox dvr/game bar, resolution, DSR and a bunch of others.