It's not normal--the plug should work in either orientation. Since it doesn't, I imagine that there is some lack of contact occurring inside the adapter, perhaps because your plug pins are a bit bent or misaligned.
If you have another adapter, try it. If not, you can plug your Type C plug directly into the Type G outlet if you first defeat the safety shutters by pressing something thin and plastic into the ground slot of the outlet. Yes, this isn't approved by the Coast Guard or the Queen, but try it anyway just to test out what's happening. Those of us who need to do this a lot sometimes use much simpler adapters that do the above slightly more elegantly:
Note that the above only applies to non-grounded plugs. Grounded plugs require a full-fledged adapter.