When you buy a car there is no icon that tells you that you have that car in your garage
This is a very well known issue. Try using the “Search” bar at the top of the page, a lot of times this can point you in the right direction for answers. A good place to start would probably be here, and definitely not in the Media section of the forums.
Forza Motorsport 5 bug report thread
Numerous people have posted about the issue there. Hope that helps!