What you're talking about is the concept of a "heavy fighter," which was a popular idea in the interim between the two World Wars.
By the beginning of WWII, most major combatants had at least one heavy fighter in development, or serving in the front line. The heavy fighter that best matches your description of a "large multi-crew plane bristling with guns" is the US Bell Airacuda, a two-engined aircraft that carried five crew and six guns. The Airacuda was designed as a long-range "bomber destroyer," to intercept and destroy enemy bombers. Unfortunately, the Airacuda was a turkey, and suffered severe technical and performance issues that prevented it from entering service. Only 13 were made, and the design had too many flaws to ever see combat.
Some examples of heavy fighters that DID see combat service include
All of these aircraft shared the same flaw when used as daytime air superiority fighters: they lacked the maneuverability and speed of lightweight, single-seat fighters. Whenever faced with combat against single-seat fighters, heavy fighters fell short. Faster and more maneuverable fighters could dictate the terms of most engagements, which left heavy fighters at a disadvantage. In addition, even the best-armed heavy fighter would have several blind spots where its armament couldn't defend it, which could be exploited by an agile opponent. Giving the fighter heavier armament and more crew to defend itself would have resulted in worse performance, making them even less capable of surviving in battle.
Perhaps the best example of these shortcomings is the Battle of Britain, when German Bf110 heavy fighters suffered severe casualties when pitted against smaller British fighters. The Bf110 was faster and more heavily armed than its opponents, and could succeed during individual engagements when skilled German pilots were able to use speed and firepower to hit fighters, then escape before the British could respond. However, when forced to escort slow-moving bombers, the Bf110's weakest traits were exacerbated, and it was unable to defend either itself or its charges.
To the best of my knowledge, the only heavy fighter to succeed as a daytime air superiority fighter was the Lockheed P-38 Lightning, a US fighter which was very successful in the Pacific theater despite being as large and heavy as any heavy fighter. However, unlike other heavy fighters, the P-38 only had one crewman, as well as some unique strengths which made it particularly suited to combat in the Pacific (exceptional range, heavy armament, high speed).
Although most heavy fighters didn't succeed at their primary mission, they had some attributes which made them attractive for other uses. Heavy fighters were fast, and better able to defend themselves than most bombers. With two engines, they could also carry an impressive load. This made them attractive as night fighters and strike aircraft, where their poor agility wasn't a handicap, while their load-carrying capacity allowed them to carry bombs, rockets, and radar. For example, although it was largely a failure as an air superiority fighter, the German Bf110 was probably the best night fighter of WW2.