Ah YES! My kind of subject! Now you're talking!
This is why that article on Streets of Rage II a while back bugged the hell outta me. RG says "name a beat em up better. You can't" or some rubbish along them lines. Well there's two off the top of my head I'd easily class as SoR II beaters personally!
Shin Nekketsu Kouha - Kunio-tachi no Banka
on the Super Famicom. Just wow. Incredible RPG style beat em up where characters develop, more come, some are killed (whoops, bit of a spoiler there!), you get more moves and stuff, you ride a bike... what more do you want?! Interesting enemies, stages... the only thing that's missing is weapons but nevertheless, Technos really did come up with some great gimmicks to solve the problem that plagued your usual 'walking along the streets' beat em up. Repetiveness.
I didn't understand Japanese when I had this (hardly do now!) but it still managed to be so engaging. And now someone has translated it!
Check it out will you?
And yes, it is one of the true Renegade sequels.
Rushin' Beat Run
again on the Super Famicom. Wonderful more tradtional scrolling beat em up with six characters, a nifty Vs. mode and great comic style graphics. Much better than the original game (Rival Turf) and loads of arcade style fun. I argued with this one against SoR at school and nobody could say anything back because they hadn't heard of it!