Tires are your number one factor in cornering. Start there. All suspension tweaks are to help tires do their task.
A sway bar has two primary functions, but with one seldom understood result. First, they try to keep a chassis level during cornering, and two, they attempt to the balance the grip between the front and rear to the drivers liking.
Both of the above are achieved by a SB transferring more cornering load to the outer tire. This additional loading of the outer tire and lowering of the load of the inner tire leads to often less total axle grip. The potential positive offset is sometimes achieved because the tire maintains better contact with the road by not leaning so much when the body rolls. That positive result however is not a given.
I'm also personally a big believer in the racing motto: "Everything effects everything else". and that starts with tires.

Let me add this and take a stab at SB visualization. When a car corners, the outer tire assumes more load than the inner tire. The outer spring compresses, and raises that end of the linked SB. However, the linked SB twists the SB and the inner end of the linked SB also raises, but it is resisted by the inner tires spring somewhat. As the inner spring is compressed, the result is less load on the inner tire, and that less load means there is less force the tire is pushing onto the road surface, and the chassis also has less force to keep it raised. and because of the cornering forces it raises less.
End result, outside tire has more load (grip), inside tire less load(grip), but is not a 1 to 1 ratio, and ultimately there is less total axle grip, but chassis feels flatter in a corner.