Ok seems pretty obvious: three stones that light up open three gates, that close after a short time. Run past the stones, use my Shout (only first level power atm) and every single time I slam into either the first gate or the second. Or use my shout first and then run... slam into the second gate every time. It's as if it's bugged. Or is it because I'm wearing heavy armor?

I walked past the 3 stones and then used my speed shout; got through the gates on my first attempt if I remember rightly.

I'm wearing a robe so cannot say whether wearing armour would slow you down or not; but I would suspect that as you are using a shout, that it shouldn't matter whether you are wearing armour or not...

Just run at normal speed (no sprint) til the third stone activates, then use whirlwind sprint, and then regular sprint the last few feet. I don't even know if you need the regular sprint, but I do it just in case.

I haven't done this in heavy armor before, but uhhh, take off your armor before you do it if that's a big concern.