I have a lot of Remington and Hornady rifle brass in 30-06 and 223.  Been thinking about getting some better brass to reload with and looking at Norma ammo.  Is the Norma brass better quality and more consistent?  I’ve never shot any and have no experience with it.

NORMA brass blows away common US brass.  It is highly consistent, has very uniform primer holes, and drilled rather than punched flash holes.  I shoot better, when I shoot NORMA brass casings.  In my experience, it is a little thicker.  Basically there are two thickness specs to .223 brass.  There is
LC/WolfGold/PMC/FC/RP/ other US Spec brass thickness (light)

and European thickness of
PPU/NORMA/MEN/GECO/Lapua

With some variability in there of course.  The European brass is thicker, which I guess is nice; but more importantly just better made brass, with better flash holes and better case-to-case consistency.

Lapua is what you want. cheaper Norma 223 is not the same brass as the more premium brass (Geco 223 IIRC).