First, connect your iPod and let it and iTunes do their sync thing. Then, within iTunes on the left hand side, click Device
On the "Summary" page, look under options to see if "Sync only checked songs and videos" and "Manually manage music and videos." On my system, these are not checked (just as a point of reference).
You'll see "Movies." Click that. Then you'll see a tick box (or check box, whatever they call it these days) labeled "Sync Movies." If this box is not checked, movies in your iTunes library will not be sent to your iPod. Checking this box and then re-syncing should allow your movies to be sent to your iPod.
I'm assuming that you'll find this box unchecked and that checking it & syncing will solve your problem. If not, then I'm out of ideas. I've never had this issue, just problems getting videos to play stutter free.
I lift things up and put them down.