Why didn't carrier based fighters support bombers?

The air campaign was already highly expensive. The Allies spent 8 billion gallons of avgas in the west alone during the war. That's almost 100 million barrels per year 1943-45, which was greater than Germany's entire fuel production. If you are going to use carriers, then this is a whole extra burden to maintain a fleet at sea year round. A carrier consumes hundreds of barrels per day. Add in escorts and support vessels and you get up into the millions of barrels per year.

The island hopping strategy in the Pacific was partly based on having access to land based air, because it is cheaper to supply. Carriers are only meant to be used in open ocean where no airbases are available. They are an extra supply cost in addition to the airplanes themselves.

Why weren't aircraft carriers used to launch fighter planes to support bombers to targets in Germany?

Because it was very dangerous to do. HMS Prince of Wales and HMS Repulse were sank because they were too close to shore.

They could have gotten the fighter planes closer. Might have solved the range problem until the mustangs could arrive.

No, it wouldn't. The range of a Seafire is 825 km. A Fairey Swordfish did slightly better at 840 km. Effectively they couldn't go any further than about 300 km. You need time and fuel to get back and remain on station. It's even worse, as those carriers had to maneuver exceptionally close to shore to give them that range. Not exceptionally close, but suicidally close.

The Seafire came into service after 1941, and the Swordfish was a torpedo bomber, not a fighter. The Blackburn Roc was retired in 1943, for lack of performance. The Gloster Sea Gladiator had a range of 800 km, and was already outdated before WW2 began.

I think the short answer is: It would have been suicide.

The North Sea is not a very large sea. And that's where the carrier fighters would have to be launched from to provide fighter support for the bombers.

In order to provide support for the bombers (which often numbered over 500), there would probably have to be at least 2 carriers maybe 3 (each with about 60 fighters). Remember you can't commit every fighter to bomber escort, as a CAP would still be necessary to protect the carrier and the escorts. So to provide a meaningful number of fighters (100+), we would need at least two very possibly 3 carriers.

With support and escort, we are talking about a LOT of high value targets in a very small amount of space, for a predictable window of time.

That is absolute suicide.

Germany had hundreds of U-boats by the time the air war really heated up.

And Germany occupied France, Norway and Denmark.

The carrier strike groups would literally be surrounded on 3 sides by enemy air fields. Worse, the Germans could come at them within 2 hours (with a flight time of less than an hour)... and a typical bomber stream could last well over 6 hours...

The odds would be heavily against the flattops.

Take a look here for information on British Aircraft Carriers in WW2. Seems like convoy protection and assisting the US in the Pacific were a higher priority to them. The land-based RAF flew night bombing missions in general.

Carrier-based fighters could have been somewhat helpful to the USAAF's day bombing campaign over Germany before the arrival of the Mustang. They could have helped with bombing the coastal area of north-east Germany, the Netherlands, Belgium and France.

However, this would have involved the USAAF, part of the US Army, admitting that their weapons and tactics for bomber defence were faulty and the US Navy could do something that they could not do for themselves. They really weren't keen on those ideas.

The Royal Air Force did send fighters along on their night bombing raids. These were twin-engine Beaufighter or Mosquito aircraft, with long range, whose purpose was to hunt and kill Luftwaffe night-fighters. They were equipped with radar and a variety of devices for detecting and homing on Luftwaffe night-fighters' radar emissions. They either flew with the bombers and hunted in their vicinity, or lurked near night-fighter bases to catch their prey at takeoff or landing. The Luftwaffe did stage "intruder" missions of this type for a while, but Hitler decided that they should defend Germany over Germany, rather than over Britain.

The only point in using carriers would be to launch the escorting fighters near Germany to provide longer range escort. That means stationing the carriers fairly close to Germany, as in closer than the UK airfields, as in the North Sea to the east of the UK. That brings those carriers within range of a large number of land based aircraft, and also puts those carriers in one of the standard submarine routes from the northern U boat bases to the Atlantic.

Consider that the Pacific theater was huge. Both sides had a large area to defend, so carriers didn't tend to face large numbers of attacking aircraft, just whatever one or two opposing carriers might be able to launch, and occasional land based aircraft.

The European theater was much more compact, so it would have not been a problem for both the Luftwaffe and Kriegsmarine to bring a large number of attackers to bear on a carrier force stationed to the east of Scotland, while also being able to reposition those forces after the attack back to their normal bases.

A similar situation arose when US forces attacked Okinawa, which was within range of land based aircraft stationed in southern Japan. The US fleet came under regular attack from land based aircraft in large numbers, and suffered accordingly. Some of the success was due to Tokko (Kamikaze) tactics, but some was the large number of land based planes that could be launched against the fleet at one time. That would have been the case with carriers operating close to continental Europe.

In the period when carrier fighter escorts might have made a difference in the ETO, which would have been 1943, German submarines were operating at their peak, and fast JU88's with torpedoes were causing significant damage to the Murmansk convoys.

Carriers close to continental Europe in 1943 would have been very difficult to defend, would have been fairly easy to locate, and probably would have suffered high casualties. Any fighter forces they held would have been hard pressed to just defend the carriers, let alone mount bomber escort missions.

Carrier-based fighters would only have been able to provide support around the north-eastern corner of Germany. The Supermarine Seafire, for example, had a range of only 825 km (this is the total flight distance from take-off to landing) at cruising speed (which was the same as a B-26). This means they only had a combat radius of about 400 km. Other planes, such as Hellcats, which had a larger combat radius would likely not have been a good match against the more agile land-based fighters the Germans would put into the air.

This would be just enough to allow the plane to fly from Cuxhaven (in the corner of the German North Sea coast) to Frankfurt, Leipzig, or Berlin. This assumes the carrier is pretty much parked on the German coastline - leaving it massively vulnerable to attack.

Also, the North Sea was an incredibly unsafe place for ships of either side - minefields and easy reach from the British and German coastlines meant that any ship operating there was easy pickings. While the North Sea was nominally under British control, major fleet operations there were avoided in World War II because of what had happened at the infamous Battle of Jutland.