By video games do you mean console games (Xbox, PlayStation, Switch, etc) or computer games (Windows, Mac)? And by “proper games” I’m assuming you mean games with a complex virtual world?
Some of the big franchises that come to mind on the console side are Metroid, Resident Evil, and Final Fantasy (and Tomb Raider, which has already been mentioned). Metroid is sort of nebulous when you actually play it, but the player character is one of video gaming’s most famous female characters, Samus Aran.
If you mean computer games, I’m only familiar with PCs, since they have the significantly larger market share compared to Macs, and there are tons of games going back to the 80s with female lead characters. If you have a Steam account, there’s Portal and Portal 2, which is like a 3D puzzle game of sorts and has the player character as female.
Also available on Steam is the Syberia franchise, which is basically an adventure game, and Mirror’s Edge, which is a first-person action game. There’s actually a whole category on Steam called “Female Protagonist” here: http://store.steampowered.com/tag/en/Female%20Protagonist/#p=0&tab=TopSellers