First, make sure you have properly installed your Sim.
Open the Launcher and click on the 'Installed Content' button. You should see the Sim you recently downloaded and installed listed under the 'Installed Content' section of the Launcher.
If you do not see the Sim you downloaded and installed, click on the 'Downloads' button in the Launcher. You should see the Sim you just downloaded. Select the Sim and then click on the 'Install' button at the bottom of the screen.
If you receive an error message stating that the Sim contains Premium Content, take a look at this article for more information about downloading and installing items with Premium Content.
If your Sim shows up in your 'Installed Content' section, then it's time for you to check if your Sim is in your game.
If you do not see your Sim, keep the following information in mind:
For example, the Sim downloaded above is an 'Adult' Sim, and not a 'Young Adult' Sim (which is the default). When you first click on the 'Custom Content' button, the Sim will not appear right-away, and thus it might look as though the Sim is not there. When you click on the age that corresponds to the Sim, however, the Sim will appear.
There is not a button to show you all Sims from all ages. You can check the age of the Sim you downloaded on the Sim's details page on The Exchange.
If after verifying that you have installed your Sim correctly, and looked in the right place while in-game, try the following steps: