There's a pretty thorough discussion of the challenges of making rear/turret gunners effective by /u/gobooks here.

The key thing here is that the faster the platform is moving the harder it is to hit targets to the side or the rear without using some sort of computer assistant. With a fighter aircraft you can simply point the aircraft where you want to shoot, and compensate for leading the target while doing so. You also get pretty direct feedback on your shots and can adjust after the fact to improve your aim. With non front facing guns you have to compensate for the movement of the target as well as the movement of the aircraft you're on, and with a fighter doing aggressive maneuvering this becomes so complicated that scoring hits becomes much less likely, vastly reducing the effectiveness of such guns. So instead it's generally always a better option to focus on maneuverability, speed, and forward facing weapons. And that's the evolution you see throughout WWII.

The very short version is that aircraft of the time were small enough that the addition of another crewman and his associated equipment was enough to reduce performance to the point that an aircraft would be unable to compete with single-engine, single-seat fighters. This was a lesson learned the hard way by both the Allies and Axis, although in different niches.

The Germans had invested a fair amount of time and effort into the concept of a heavy fighter as war loomed, culminating in the Messerschmitt Bf-110. While the Bf-110 on paper offered impressive performance, reality soon set in once it went up against comparable air forces and it was almost helpless against single-engined fighters - so much so that Bf-110s were often themselves given fighter escort during the Battle of Britain, and for much of the rest of the war they were relegated to roles where they weren't intended to go up against fighter opposition.

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On the other side of the English Channel, however, the British had taken this concept to what sounds like what you're talking about - an otherwise conventional fighter, but with a rear gunner. This concept had existed in the Royal Air Force in some form since the First World War, where two-seater aircraft performed reasonably effectively. Though these aircraft were nominally bombers, the RAF would continue to operate some as two-seat fighters. The predecessor to the British turret fighters of WW2, for example, was the Hawker Demon. The Demon seized on the excellent performance of the Hawker Hart light bomber and optimized it for use as a fighter - adding an extra machinegun for the pilot and improving the field of fire for the gunner. About 300 Hawker Demons would be procured, and in addition to using their gunners for defensive fire, they experimented with coordinated attacks using the gunners against bombers.

These experiments with the Demon in the mid 1930s would lead to the development of dual requirements for the RAF and Fleet Air Arm for turret-armed fighters. The RAF requirement - which produced the Boulton-Paul Defiant - was focused on the role described above, with the turret serving as a tool for dealing with bombers as well as a means of defense. The Royal Navy requirement was less concerned with bombers and more with observation in mind - the second crewman would increase the aircraft's situational awareness, reduce pilot workload for navigation, and make them more useful for reconnaissance.

So when World War 2 broke out, the RAF was operating the Boulton-Paul Defiant and the Fleet Air Arm was flying the Blackburn Roc. The Defiant by this point was already known to be unable to deal with enemy fighters, as tests soon after it entered service would show. Nevertheless, they'd see some limited early successes as German pilots unfamiliar with the type fell prey to its four-gun rear turret. Crews would do their best to develop tactics to maximize their defensive fire, but heavy losses during the Battle of Britain would see them withdrawn from daylight fighter roles entirely and relegated to night interception - where fighters would not be encountered.

The Roc was similarly underwhelming due to the heavy turret it mounted, but it had more longevity. Nevertheless, they were overshadowed by the more lightly armed Skua that they had been developed from. The Skua was more analogous to the American Dauntless dive bombers, being a multirole light bomber with limited defensive armament. However, the niche that allowed the Fleet Air Arm's two-seater fighters to persist through the end of the war was one that made the rear gunner all but irrelevant in the first place. Fleet Air Arm operations more often than not took place in an environment where enemy fighters were few and far between - when aircraft were encountered, it would be either unescorted bombers or lone patrol aircraft that a two-seater like the Skua or even Roc was perfectly capable of dispatching.

The Fleet Air Arm would implicitly recognize the uselessness of the defensive gun as it operated its later two-seat fighters - first the Fairey Fulmar and then the Firefly. The Fulmar had the ability to mount a single defensive gun, but more often than not it was left on the ground to save weight. The Firefly did away with the defensive gun entirely, but retained the second crewman that aided in observation and navigation.