If you own the game legally, Here is the official fix:
There are five common reasons why you might see the missing steam-api.dll error in Tomb Raider 2013 : Tomb Raider 2013 Steam-api.dll Download
These errors typically appear immediately after launching the game, before the main menu appears. If you own the game legally, Here is
You’ve just installed the iconic 2013 reboot of Tomb Raider . You’re ready to guide a young Lara Croft through the treacherous Dragon’s Triangle. You hit the “Play” button, and instead of the dramatic opening sequence, a pop-up window shatters your excitement: You hit the “Play” button, and instead of
A brief dip in your internet connection during the download can lead to a partial or "broken" file.
This is the safest and most effective solution. Steam will compare your local files with the official versions and replace missing or corrupted files, including Steam-api.dll .