For the longest time, British lawyers weren't allowed to do cases on a contingency fee basis. Then about twenty years ago, the rules changed and they *were* allowed to do it -- initially just for personal injury, but eventually for all civil litigation -- but it's extremely rare for the law firms themselves to fund the case.
The U.S. was like this for some time, too, under the same ancient doctrine known as champerty, which prohibited lawyers from owning part of a lawsuit where they were representing another party. At least in the case of British lawyers, it was because a lot of old money had basically cheated poor people out of their property generations ago, and lawyers start representing them for a cut of the potential profits, but worse, they started winning.
It was dressed up in some nonsense about ethics, but it was never really about that.
Some vestiges of the doctrine still apply in contract law with contracts that explicitly allow a lawyer to "own" part of a lawsuit, but largely, the doctrine is dead.
In some places, advancing costs is still prohibited under a related doctrine called maintenance.
It's a family of three general doctrines, champerty, barratry, and maintenance, with the second being the only inherently unethical one, which is going around stirring up litigation between third parties. Even that is somewhat tolerated with activist legal groups both on the left and right seeking out sympathetic test cases or, on the less ethical end, hiring a "Slippin' Jimmy" type to go around looking for even the most trivial ADA violations to make a case to sue about.
"Fake bill" isn't an accurate way of describing it; the recovery is for "reasonable attorney's fees," so you submit what would have been your billed hours and rate if you were doing standard billing, and the judge determines whether or not those were reasonable and adjusts accordingly.
There's generally what's called a "lodestar analysis," related to
quantum meruit ("the amount one deserves"), that is, the amount a reasonable attorney would charge. This usually requires actually itemizing the billing so the court can analyze whether or not these actually are reasonable fees. Sometimes something so nonsensical as actually to constitute a "fake bill" does get past a judge, but the usual signs of such billing are block billing that doesn't really identify what the fees were for, with some vague designation like "legal research," inflated fees either outside the realm of what is normally charged in the area, or rates that are clearly inflated for the level of skill of the attorney.
The billing for Mosquito Boy's work in Vic's case would be a perfect example, and claiming to be a total expert on TCPA while billing as many hours for research as Sam Johnson, who never purported to be an expert, would be an example of a nonsensical bill. This was the one thing Chupp got half right and so of course is the one thing the dipshit appeals panel remanded.