Christopher, most directions I have seen for tuning carbs tell you to do it only after the mechanics are known to be good and especially after verifying that the timing is right. I'm going to assume that the engine is in good shape mechanically and that it is fired by a magneto which means there isn't any timing to worry about.
The next step is to turn the idle speed screw in so that the engine is starting to run off the main jets, not just the idle circuit. For a lawnmower engine, we're probably talking 1200 RPM or so, that is, a "fast" idle or slow "run" condition. Then, turn the mixture adjusting screw out until the engine starts to miss, or bog down, or the RPM drops even a little bit. Note the position of the screw, that is, turned straight up at 12:00 & 6:00 or horizontal at 3:00 & 9:00 or somewhere inbetween. Then start turning the screw in 1/2 revolution (from 12:00 to 6:00 etc.) at a time, pausing just a bit at each 1/2 turn to let the carburetor settle out a bit. You can also rev it up a bit to clear it out as you move the screw in. Keep turning the screw in until you again detect a miss, some stumbling, or any loss of RPM. Then back the screw out half the distance you screwed it in. You can repeat the process until you are satisfied you have found the "happy medium."
You can also start by turning the screw in until it stumbles and then back it out until it drops RPM and then screw it back in half the way to the "happy medium."
Some carburetors and engines are easier to tune than others. Most folks feel that a little rich is better than a little lean, so they will run 1/4 or 1/2 turn rich from the "happy medium." I used to try running a little lean, which it sounds you like to do, but fixing burnt valves and warped heads cured me; now I like to run just a tad rich.