Treat the players who come first and second as one person, and third and fourth as one person, and follow a normal tournament structure. So first and second are one side of the bracket for the next round, and third and fourth drop down to the loser's bracket or out entirely depending on your model.
If your number of players isn't divisible by 4, that's OK, just have a 3-player or 2-player match with the people left over. You could choose to skip two-player matchups and automatically put them through to the next round, or match them up and put through only the winner. Obviously, the deciding match has to be a 2-player match.
For example, single elimination with 7 players: A, B, C, D, E, F, and G.
A,B,C,D play and E,F,G play. A&C win and F&G win and all play each other. A&G win and play each other. A triumphs.