So l have been snowboarding three times so far and l can link turns (from toeside to heelside). l know that is not carving quite yet since you need to be riding on the edge at all times for that to occur and l am not doing that yet. The reason is that l am scared to go fast so l try to skid and slow myself down. Then l turn my board like this ------- and start again. How do l improve and learn to carve and also be able to go down steeper slopes? Also, what is crossing-over and crossing-under?

Go fast but always stay on edge, never ride flat based when going fast. If you are always on the edges of the board carving is easy and you can be confident that you wo not catch an edge and fall.

l have no idea what crossing-over & crossing-under mean.
My advice would be to just go for it. It does take some courage to
build up speed & make ur transitions from edge to edge.
Start on a smaller slope (but not the bunny slope) & go to an area that
is least crowded. Get some speed up & then make ur cut.
If u fall, u fall. It is a part of the learning process.
But like the other person said, u really want to stay on ur edges
and avoid riding flat. Just start going down the slope & switch from
heel-edge to toe-edge. As u pick up speed, make wider transitions & u will be carving in no time.