To answer/comment on everybody at once, with the proviso that I always approach Game Problems from the aspect of What Was The Historical Reality and Why Isn't It Happening - or, Is It Happening And We Don't Like It.
First, the historical counter to Bombers was never AA, it was Intercepting Fighters. A combination of the two was even better, but the bulk of Bomber Losses and the real deterrent to Bomber Attacks were Fighters shooting them down in large numbers. That was true from the "Zeppelin Blitz" of 1917 - 1918 through the "London Blitz" of 1940 and the Luftwaffe attacks on Moscow in 1941 - Moscow had more AA than London, but when the fighters of the 6th Air Defense Fighter Corps began intercepting the bombers, they stopped attacking. The Royal Air Force's Bomber Command and the USAF 8th Air Force both had to stop bombing German cities temporarily in 1943 because the German Fighter Defenses were inflicting too many losses - this even though the Germans invested almost 1,000,000 men and thousands of guns, searchlights, and radars in their AA defenses. The AA made Bomber Attacks exciting, the Fighters made them suicidal.
Second, Bombers are expensive. To do serious damage to a City - or a military force the size of Units in the game, which we have to assume are at least Divisions (10,000 - 25,000 men each) takes not dozens, but hundreds of bomber aircraft. Each one costs 2 - 5 times more to produce in resources and Gold than a single-engined Fighter, and has an expensively-trained crew of 3 - 12 compared to the single pilot in the Fighter. In other words, a large Bomber Force, even discounting all the extra Command and Control and Infrastructure required to make it effective (see my Post above) is very expensive to Build and to Maintain. The German Luftwaffe tried to do it "on the cheap" by using almost exclusively 2-engined medium bombers for everything and even (in Game Terms) converted Fighter Units (Stukas, Fw-190 Fighter-Bombers) and they were utterly ineffective against Cities and Infrastructure on the scale of the game (that is, they could knock out a Bridge, but not a District-wide Industrial/Commercial Zone or Railroad).
Third, there is a big difference in effectiveness of Bombers versus Big Fixed Fat Targets like Cities, and Dispersed Hiding Targets like military units. The Luftwaffe's Converted Fighter Units were extremely effective against military units in 1939 - 1942, a waste of their cost against cities. Heavy Bomber units were, by 1944 at least, City Killers: see the effects on Dresden or Berlin before the Red Army even got to them, or the destruction of Tokyo in 1945 by 'Conventional' Bombing that killed more people than both Atomic Bombs combined. On the other hand, it took a concentration of Heavy Bombers over Normandy twice the size of the average force used in a raid on a city to seriously degrade a single Panzer Division. Basically, effectiveness of Bomber Unit against a Ground Unit is half or less the % damage the Bombers can inflict on a City or District - and the District or City can frequently be repaired unless you keep right on bombing it.
So, I would contend that the answer to the OP Bomber is:
1. Jack up the Anti-Air Factor of the Fighter versus the Bomber. The +10 suggested might be enough,
@Sostratus could undoubtedly tell us precisely what Fator Differential is required to make Fighter versus Bomber attacks virtually suicidal for the Bomber.
2. Jack up the cost of both building and maintaining Bombers. I might even go so far as to require a regular Aerodrome District to launch the sort of concentrated Bomber attacks required to seriously degrade Cities or Districts.
3. Change the Promotion Tree of Fighters so that Strafe and Tank Buster came as 1st and 2nd promotions instead of 2nd and 3rd, and get rid of Close Air Support Promotion for Bombers - that really effects All air units and should probably be a Military Civic in the late Modern or early Atomic Eras. Replace Close Air Support with Pathfinders = + Strength against City Center, District tiles or Improvements. That would more effectively show the real power of massed Bombers against the Big Fixed Targets, and the fact that you have to concentrate resources on either attacking Units or Cities, because the same air units cannot do both effectively.
4. (If necessary) have an 'automatic' Repair of Bomb Damage to City Centers and Districts, which can be increased by Investment, like a Military Civic of Civil Defense in the Atomic Era that gives an extra % recovery every turn.
