Best cost-effective graphics card with good OpenGL integration?
 

In the past year or so, I have been progressively been moving from Windoze to Linux. After much looking, I have chose Linux Mint 13 KDE as my main distro for now. But that doesn't matter.

What does matter is that my ATI Radeon HD 5770 graphics card has horrible OpenGL support. My games played fine in Windoze, with DirectX, but whenever I play games with OpenGL, it's slow as hell. In Windoze, I could play most games on medium-high graphics with a solid 60 frames. When I try to play the new Team Fortress 2 beta for Linux through the new Steam for Linux beta, even with all of my settings down, max fps I can get is around 40, and it's not consistent at all. My frames drop a lot whenever I do anything at all.

I do realise that my GPU is becoming outdated, so what I really want is a good (relatively cheap) GPU that works well with OpenGL. Does anyone have any suggestions? Thank you for your time.

No need for a new videocard, I would think, that card should be fast enough for TF2, as long as you use the proprietary AMD drivers, not the (installed by default) open source drivers.
Which drivers are you using?

I'm using the proprietary drivers. And I wouldn't just be playing TF2. I bought a lot of games on Steam when I played on Windoze, and would like to play them when they're released on the Linux version of steam. And since OpenGL is what Linux USES per-say, I would like to have good OpenGL support on my GPU anyways.

You have good OpenGL support on your GPU. If you have problems with speed I doubt that this is because of a slow CPU (TF2 is a Source engine game and your GPU should have no problems at all with handling the Source engine) or lacking OpenGL support (the HD5770 supports OpenGL 4.3 with the proprietary drivers, the newest openGL version).
There seems to be a different problem with your system's setup that is slowing you down (for example, unity is known to cause some performance issues).
Of course you can feel free to buy a newer videocard, it is just that i doubt that this will solve your problems.

Then why do games run much slower under OpenGL? And the only other problem I could see is running KDE, but it's actually shown to be better with running games than Unity is. Is it because I run a dual-monitor setup and more GPU is used?