Don't think so, I managed to complete that mission way earlier that the recommended level.
Are you absolutely certain you are following the correct route?
yes, I followed it on the minimap. Do you jump when the barrels come at you?
Is this race bugged?
Definitively no: https://www.youtube.com/watch?v=XDyRcRuTKCA
Do (or did) you use any cheats, mods, scripts etc.?
no cheats or mods used. Maybe if my horse leaves the path slightly, it will fail