These two paragraphs say it all:
“There is little that the regulators could do, except for waiting for the new entry in response to the profits currently being made in this market,” says Bilotkach. What might help, then, is if U.S.’s airline industry resembled Europe’s more closely. In Europe, low-cost carriers such as Ryanair nip at the heels of large incumbents such as British Airways, making ticket prices more sensitive to major drops in operating costs. There, airlines’ margins are about half what they are in the U.S.
But it’s relatively hard for new entrants to get a foothold in the U.S.: Market share is hard to accumulate when takeoff and landing slots at major airports are in such short supply. And besides, when it comes to regulation, as The Economist pointed out, “having approved the wave of mergers, it would be a bit rich for officials to complain that there are not enough big airlines to ensure competition.” So for now it’s unlikely that airline executives will do much to lower ticket prices.
Regulation/Regulators can't fix the problem by adding more regulation, in fact they probably make it worse by increasing the cost barriers to entering the market.
Want more competition? decrease the entry barriers. Expand the slot controlled airports. Eliminate taxes/fees on startup airlines for the first "x" years of operation. Etc etc.
This country used to have hundreds of airlines, almost anyone with a few (possibly borrowed) nickels to rub together could start one and would have a shot at competing. Recreate that environment, use it to replace the one that favors mega airlines, and the competition will come.
And General Aviation flying is still expensive because 100LL prices have hardly dropped $1/gallon despite the fact that automotive gasoline is sitting at $1.50~$1.70 in many areas.