Prince Jackson -- The Girl Is Mine ... After Pricey Dinner Date

Prince Jackson stepped up his royal pimp game -- taking his Kuwaiti princess girlfriend out for a fancy Hollywood dinner ... and all signs point to the princess being impressed.

Michael Jackson's 16-year-old son and Remi Alfalah had dinner Friday night at Madeo -- some of the best Italian grub in L.A. -- and judging by the grin on her face ... she gave the date 2 royal thumbs up.

A couple weeks ago Prince and Remi were spotted on another date -- but that was just daytime bowling.

The jump to a high-priced dinner is a clear sign Prince is making, y'know ... progress.