What type of paint can be used over oil-based primer?
If we're talking about a "modern" oil-base like Kilz or Cover Stain, I'd say you're on the right track. Oil-based primer offers the best adhesion for your paint.


  My kitchen gets very steamy a lot.


My bet is you have a flat latex paint on the walls. I would repaint with a semi-gloss enamel (can be latex). The semi-gloss will resist the moisture better than flat paint.