There are tons of ways to do just about every knitting technique. The other day someone posted about a type of cast on I'd never heard of.
Anyway, if you are following the videos for M1L and M1R, you shouldn't end up with a gap, but one thing you could try is knitting into the back
of the stitch when you do your M1. This will twist the stitch and make it tighter. I usually do this when I'm doing a M1 on the purl side to avoid the gap.
You could try some of the other M1s too to see if they work better.
One other question--are you picking up stitches? If so, it might be that you need to pick up a few extra stitches. You can do this, you may just need to adjust the pattern a bit for the thumb.