All fighter planes, since the bi-planes of WWI, have been designed with the primary purpose in mind of destroying enemy aircraft and claiming air superiority in a given area.
There have been a number of fighter aircraft which have broken from the tradition of being maneuverable, in favor of trying to engage enemy aircraft exclusively from beyond visual range with longer ranged missiles. And for a short period of time in the early 60's the USAF design philosophy favored this approach. But throughout most of the past 90 years of air to air combat, "most" fighter aircraft have been designed with a great emphasis on maneuverability.
The F-4phantom was originally designed as an all purpose aircraft, fighter, ground attack, recon, jamming station, etc. and useable from aircraft carriers as well as traditional runways. It's speed, range, payload, all weather capability, etc. were a huge improvement over previous generations of fighter aircraft. Early models were more maneuverable than many previous planes like the F-104, but slightly less maneuverable than something like the F-86 which was built from the ground up for dogfighting. Entering Vietnam with the F-4's there were two problems. 1) the airforce was really pushing this idea of medium & long range missile based fighters which don't need to dogfight. And 2) the airforce had almost no air to air combat training (these problems were related).
In vietnam it quickly became apparent that this philosophy was flawed. So changes were made, both to the F-4, and to the pilot training in the air force. The F-4E was equipped with the now standard 20mm gatling cannon, and had increased flap area, and two additional weapons hardpoints to carry yet more ordinance. It also had leading edge slats added to the wings which greatly improved the plane's maneuverability and resistance to stalls. Improvements were made to the AIM9 sidewinder short range air to air missle, and the existing F-4D models were retrofitted to be able to use sidewinder missiles as well, instead of the horrendously worthless AIM4 Falcon missiles it originally came with. The airforce and navy both instituted elaborate combat training programs for their fighter pilots, the air force at Nellis, and the navy created TopGun. Both taught primarily by ace pilots of previous wars. Between the gun, beefed up handling, and probably more than anything the advanced air combat training, the F4 was doing MUCH better towards the end of the war.
The enemy pilots were not inferior, they were actually very well trained by the USSR. And they took full advantage of our 'rules of engagement' in vietnam to make sure they could always close to dogfight distances at the start of an engagement, catering to the mig's strengths.
F-4's will happily pull enough G's to knock the pilot **61**, like most modern fighters, and can perform any conventional modern jet fighter maneuver which doesn't require a thrust to weight ratio greater than 1:1. The F-4 models E and up (the models go to like S or so) are not that much less maneuverable than the F-16C (at least with the pratt & whitney F100 engine it came with originally).
Worth noting, the J-79 engines in the F4 phantom are not 'huge' by any stretch. On the large side for the time, but not 'huge'. Granted, using two of them did provide more power than anything else flying at the time. They were used in previous jets though. Compared to modern US jet fighters, only the F-16 has less thrust than an F-4. The F-18's engines are slightly smaller, but have a bit more thrust than the j-79's. It was a hot rod by 60's standards, but actually a little slow by 80's standards.
The F-4 was by the way the first plane to dabble in the idea of using a primitive form of "vectored thrust" to increase maneuverability. the rear horizontal stabilizers are one piece titanium skinned flaps, which actually dip into the jet exhaust at high angles of attack, giving the plane a bit more turning power. That's why there's a section of rear horizontal fins which don't get any paint.
I'll grant that the original F-4's (C & D models) had some pretty obvious growing pains, they were good all-around aircraft but bad dogfighters. But on the other side, i'd also argue that the F-4E was one of the first fighters in the world to refine a very good balance between the stability you need for high supersonic flight, with the maneuverability you need to win a subsonic dogfight, and the thrust to do it better than slower dogfight-specific fighters.
If you want huge engines, the F-15 carries the largest (the same as the F-16, but two of them). Technically the engines in the f22 have more thrust, though they are smaller than the f-15's engines.
Anyway, extremely powerful engines are actually a great help to maneuverability. Having more thrust than weight allows a fighter plane to perform a wider range of maneuvers. Some maneuvers you just can't execute without more thrust than plane weight. Additionally, the ability to quickly regain speed lost to elaborate turning maneuvers is also a huge benefit to fighter planes. As is the ability to rapidly gain lost altitude. The F-15 is the most maneuverable plane in the US inventory in part 'because' it has an absurdly huge amount of thrust. And in part because the airframe is super stable and stall resistant, and designed to withstand enough G forces to kill the pilot several times over without cracking.
As far as W/L ratios, it's worth noting that arabic fighter pilots tend to suck, really badly. It's not a racist observation, and it's not their fault really. It's just that arabic governments have a very very strong tendency to completely fail to give them any training beyond basic flight school. Relying on kismet to be victorious hasn't worked out so well for them. Also the export models of aircraft they buy are generally 1000-2000 lbs lighter, because they lack all the cool electronics that would make the planes capable in a modern war. Literally 2 or 3 AWACS planes, or twice that many EA-6's, could make every unfriendly aircraft in a place the size of iraq completely blind on radar, and largely unable to radio communicate. It's pretty unfair. Israel's planes are similarly "stripped down", but they have home-made electronics suites (mostly designs stolen from the US) ready and waiting to be installed when they take ownership of any new planes, giving them a massive advantage over the electronically stripped down aircraft of their neighbors. But i'm sure the f-15 would kick @ss very nicely, even if it wasn't playing on a heavily slanted playing field in most of it's actual combat time.
Here's a modern marvel that they only made one of:
STOL testbed, MTD testbed, F-15E avionics testbed, ACTIVE testbed, and finally adopted by nasa.
It would have been an amazing dogfighter. On the hush hush, it could drop 500mph in seconds, fly at an 85 degree angle of attack, basically it could not be stalled, because it's movements both in and out of stall were controlled and intentional. So it could use stalls, on purpose, just to turn around fast. And it could pitch the nose of the plane away from the plane's actual path of movement, to shoot at other aircraft with the cannon, basically shooting across the inside of a turning path, and use the radar to automatically help guide the cannon's burst onto the target (a la phalanx cannon). and take off & land on runways a fraction of the length needed by the F-15.
Ironically russia liked it even better than the US, and it inspired a pretty cool new model of SU-27/35 called the SU-37:
even without a cold war, they're pretty tight lipped about it's abilities, and don't send them to air shows.