As an American, my favorite Turrican game is Super Turrican; it's an exceptional run & gun. Second would be Mega Turrican. After that I'm not sure, but it'd be Turrican 2/Universal Soldier and Super Turrican 2. All four of those games are A-grade classics for sure, they're great. As for the first game though, it's okay but frustrating and not nearly as good as the later ones thanks to blind jumps, some design issues, etc. The later games improve hugely over it.
I own every Turrican game
Even the shitastic Genesis games by Accolade (Universal Soldier and Turrican)
Yes, they have their issues, but you're being way too hard on the Genesis versions of the first two games. Sure, they're downgraded from the Amiga in a lot of ways, but they're competent versions of the games. Comparing the console versions of the first Turrican, the TurboGrafx-16 and Game Boy versions both cut some levels out, but the Genesis version is complete, for one thing. Of the console versions of Turrican, the Genesis version is clearly the best one. The graphics and sound are better on the Genesis than the other two versions as well. And as much as the music is definitely worse, at least you get better controls with multiple buttons. But on the other hand, am I remembering right that they made you lose health way faster on consoles? If so that's pretty annoying.
As for Universal Soldier, obviously it's much better on Genesis than Game Boy, comparing the two released ports. The unreleased SNES version is about the same though I think. But on the Genesis, the game has seriously downgraded music versus the amazing Amiga soundtrack, but it still sounds good. They do also cut the shmup levels, which is too bad, but the new levels that replace them are solidly fun, so that at least partially makes up for it; they didn't just cut stuff without adding anything new. And sure, the graphical overhaul some of the sprites got is silly, but it's the same game underneath that so that doesn't bother me. And it adds password saving to the game too, which is great. Universal Soldier is the only Turrican game with saving, and more should have had it too.
Turrican 3 is better than Mega Turrican though, but only slightly
Better sound on Amiga v. better graphics and controls on Genesis. Easy Genesis win there, up to jump is the worst!