Keihin PJ carbs are crap for bottom end power (this is what the 125RE has?). They're basically a methanol or road race carb - the design clearly favours big throttle openings and lots of fuel flow (compared to, say, a PWK). My 93 CR125 was basically untuneable with the PJ - wouldn't idle and had awful low throttle response.
Swapping to a PWK has massively increased the bottom and midrange power, and it starts more easily and idles properly. No loss of top end.
FWIW, I've been finding a lot of my older bikes have required significant leaning off to run properly - it really makes me wonder how these things get to 20++ years of age with such crappy jetting... I'm beginning to suspect that modern fuels are significantly different, or something.
Eg: 88 KX250 went from 52/172 to 45/162 and is still slightly too rich on the pilot.
Anyhow, how many turns out is the idle mixture screw? More than 2 full turns and you should look at a smaller pilot.
Another giveaway of an too large pilot jet, is if the idle speed screw is wound in a long way to get it to idle (and the idle mixture screw doesn't do much).