Here are the states that voted on or before Super Tuesday in 2008 (when McCain formally secured enough delegates for the nomination), shaded by which party had more voters in its primary:
MI & FL were special cases (disqualified DNC delegates); no battlegrounds out of place otherwise. However, there were multiple examples where Democrats vastly over-performed in terms of turnout in states where they had no business being in the majority (not to be confused with Dixiecrat state/local primaries).
There's little to no connection between having the higher turnout in a primary and winning the general. The Republicans are excited like the Democrats were in 2008 after eight years of rule by the other party. Democrats, on the other hand, have been told to expect a coronation for the better part of a decade. It's not shocking at all to see Republican turnout thus far be higher in states where there are (roughly) equal numbers of Ds and Rs.
Are you sure about Illinois? I thought Democrats shattered turnout records that year.
Anyway, the New Hampshire primary was contested for Republicans (at least for second) while Dems had a blowout. I'd have to guess that an overwhelming majority of independents went with the GOP ballot this time.
