No, that shouldn't happen.
First, take a look at your "Account" tab, it should show in color all the boxes of the expansions you are currently subscribed to (grey are the expansions you don't currently subscribe).
If it shows that you don't have the expansions that you think you bought, then something went wrong with the buying process and we need to look there.
If you actually own the cards, then it may be that you didn't set those new cards as familiar in the "Familiar Cards" tab. If you do that, they will appear in games that you start on a table or vs. a bot. An easy and fast way to do that would be to click on the "All expansions" button, which makes all cards familiar to you.
If all of this doesn't work, you need to ask again, and described as detailed as possible how you start your games: Do you make a table on the "New Table" tab, or do you join other tables or do you click on any (which?) button in the "Matching" tab.