We (RAF) used to carry out "fighter evasion" exercises and probably still do, although the exercise name changed to "fighter affiliation" at a later date. There was no FJ fighter that could turn tighter than a helicopter in the hover, or flying at very low speed. Helicopters can also stop and hide behind things...
Some F-4 pilots (who usually couldn't even see us to take a shot, whereas their smoky exhaust trails made them stand out miles away) used to claim that if they couldn't shoot us down, they would attempt to "knock us out of the air" by overflying low level at Mach 1. Not sure how they could do that when they didn't even see us. We could, of course, land anyway...and an armed helicopter could still shoot upwards.
Most experienced Harrier pilots I spoke to said that they probably wouldn't waste time and fuel trying to dog fight with a helicopter as an opportunistic target and if they knew it was likely to have a shoot back capability they would not risk getting mixed up with it.
A slow speed, highly manoeuvrable turboprop aircraft would have more chance of a shoot, hence the Pucara. The Warsaw Pact also developed the Frogfoot, possibly also a different kettle of fish, especially if the pilots were well trained/experienced and able to hunt in packs.