"The display driver stopped working and has recovered"?

That's a display driver crash and recovery. Try completely uninstalling all your display drivers (so the screen reverts to 800x600), then reinstall the latest driver.

As I said, what you described is a display driver crash and recovery.

I suggested completely uninstalling and reinstalling your display driver because that's easy to try and costs you nothing. Yes the driver crashes may be due to a hardware problem. But those are hard to diagnose and expensive to fix, while trying a software fix is free and relatively easy. So you should assume it's a software problem first, and only after you've exhausted software solutions should you conclude it's a hardware problem.

Sometimes some cruft from an older driver gets left in place which interferes with newer drivers and causes crashes. This is especially true if you've changed video cards, which it sounds like you've done. Sometimes driver configurations which worked with the old card don't work with the new card. Completely uninstalling then reinstalling the drivers is the best way to clear that up (short of a complete reinstall of your OS). If that doesn't work, the next thing I'd suggest is changing your browser's settings so it'll do video rendering in software (you might have bad video codecs installed), since you say the problem happens most frequently when browsing, and video playback is the strongest interaction between your browser and video card drivers.

Using an adapter to go from 4-pin to 6-pin or 8-pin generally isn't a problem. The wires all go back to the same voltage rails. The yellow wire on the 4-pin molex is 12V, the red is 5V, and the black wires are ground. The 8-pin molex is 4 12V wires and 4 ground wires. By using a 2x4-pin molex to 8-pin adapter, the only difference from a "native" 8-pin adapter is you've taken two 12V wires and converted them to four 12V wires.

The extra 12V wires on the 8-pin are used to reduce the voltage drop and heat generation (in the wires) due to resistance. Thinner or fewer wires = more resistance = more voltage drop and more heat. While a voltage drop can affect your card's stability which could be causing a driver crash, you'd expect to see that more during heavy gaming when the power draw is highest (voltage drop is proportional to the power draw). Not during web browsing. So your symptoms do not fit the PSU wiring being at fault as you've concluded.

Is that a better explanation of why my original suggestion was to completely uninstall then reinstall the drivers. I don't always have time to write up a 4 paragraph explanation.