I mean, i'm going for Onyx medals and seriously 3.0, so I play about 85% Private matches trying to get the medals and 15% versus for fun. In short, the Onyx Medals give you a good amount of experience, but if you aren't going for seriously 3.0 and only want to rank up quicker Ranked TDM is probably your best bet.
It all depends on what medal you are going for, when I was doing the pyro kill onyx and getting 130 kills in a 12 minute (150 to cap, 12 minute KOTH match) I was only getting 2k experience a match. But when I was doing leader saves for example a 12 minute match would give me like 300-400 xp, if even that. Then there are some medals like founder where you legitly get about 0 experience while going for and take 7-8 hours to get.