This one is more of an overall action game, but half the stages are beat 'em up. Mad City has a fan translation here: http://www.romhacking.net/translations/1554/
So you can play that version, which differs a lot. For starters, enemies are much less durable than they are in Bayou Billy. You'll even start off with green clothed enemies, that take fewer hits than any of the characters in Bayou Billy. You don't even need to actually fight with crocodiles, and the swamps they reside in have land masses to help you avoid them.
Select: Switch between punch/other weapon and using a gun, when you get one.
Kick's have a longer range, and are stronger than punches. A & B jump kicks. That's all there really is to the beat 'em up segment controls. You're health is measured in increments of 7 red square blocks. Weapons and items fall out of beaten enemies, and are collected by walking over them. These include sticks, knives, guns, bullet proof armor, whips, & meat(restores health). If you're on the same continue(of which you have infinite until the final stage, which is instant game over, if you lose your last life), you'll start roughly where you were when defeated. When you have a gun and press select in Bayou Billy, the bullets count will flash, letting you know you're about to use the gun with the B button. The first and 3rd stages have you in water at times, where you can't kick, but can jump kick. One section in each of those stages of Bayou Billy requires you to fight off 2 crocodiles, otherwise the game won't move forward, which is actually easier to do by being in the water(One in stage 1 drops a stick, so if you can get it before it picks it back up, fighting them will be a bit easier. In stage 3, an earlier enemy has a stick, so it'd be easier if you can keep it by the time you need to fight the crocodiles there, which both drop meat). One odd aspect is that the game won't give you any real clue as to when you can move on, except that enemies are no longer coming at you. This means you have to move to the right edge of the screen yourself, to make the screen scroll.
There are first person shooter/light gun stages, depending on which mode you choose. Mode A in Bayou Billy, and Mode B in Mad City require the Zapper, whereas Mode B in Bayou Billy and Mode A in Mad City allow you to play with the pad. If you're using the D-pad, either button shoots, otherwise the Zapper, plugged into the 2nd player port, should be easy enough to figure out(in emulation it'd be the mouse, except the "Zapper" doesn't seem to be recognized in a patched Mad City). Bayou Billy will tell you which you'll be using before every stage. Sometimes enemies in these stages drop items, which you must shoot to collect. The effects of these include extra ammo(kind of looks like a gun barrel), temporary infinite ammo(hourglass), temporary invincibility(vest), instant screen clear(star), an extra life(a little man), and health restoration(health kit, white box with a red cross). Running out of bullets in these stages will automatically cost you a life, and there's a boss battle to contend with in both. Try to avoid wasting bullets on enemies that won't stop to shoot you, especially in Bayou Billy, where you have less ammo than in Mad City(you get even less ammo in Zapper mode). Bikers in the 2nd shooting stage will always leave behind an item when defeated, mostly ammo.
There are driving stages as well. During these, Up gets moving, Down slows down/stops, Left/Right steers to avoid other vehicles(you're instantly destroyed during a collision), B shoots, which takes out enemies in front of you(except the one you're chasing the whole time in Mad City), and A throws grenades(BB)/dynamite(MC), which takes out air vehicles, that try to drop bombs on you. There are cans in the road that can be picked to regain health in Mad City. The object seems to be just to make it a certain distance. Bayou Billy adds obstacles on the side, and a bar at the top that shows your position between starting and ending points. In Mad city you randomly have puddles of water, or oil, to slow you down for a split second. Bayou Billy still includes them, but also adds large rocks in the middle of the road, to instantly kill you. Bayou Billy also has a lot more turns in the much longer roads you need to travel, while it still counts down with a timer. These stages have no bosses.
The stage layout looks like this:
Stage 1: Beat 'em up (no boss)
Stage 2: Shooting
Stage 3: Beat 'em up
Stage 4: Driving
Stage 5: Driving
Stage 6: Beat 'em up (no boss)
Stage 7: Shooting
Stage 8: Beat 'em up
Stage 9(final): Beat 'em up
Mad City has a bad ending that requires you to, after you rescue her at the very end, avoid Annabelle for a while.
Speaking of, one awkward bit of censorship in Bayou Billy is that Annabelle, who has a red dress in Mad City, is wearing a low red cutoff shirt, and jean shorts. In this case, she's showing off much more in the North American release.