burnout paradise hands down. by far the best racer on xbox360 imo.
need for speed most wanted (the launch title, not the new one) and burnout revenge were also very very good, but those are like year 1 360 titles so the graphics aren't that great, but they are still really fun. i played burnout revenge within the past 2-3 months and it is still a blast. the crash mode in it is a lot of fun to this day.
many of the NFS titles in general are pretty fun, but they are all so similar. hot pursuit was my favorite nfs game other than the nfs:mw launch title.
of recent, forza horizon is really good too. this is more what you seem to be looking for. it's more of a hybrid style. the burnout and nfs series are flat out arcade racers where you can make hairpin turns going nearly 200mph. horizon is between forza and an arcade racer, and it's open world.