In order for you to turn your question into a thesis, you need to choose one side of it. For example, you might state the thesis as, "The subplot in King Lear strengthens the main plot". To have a good thesis statement, you need to be able to take one of the major themes, conflicts, elements of plot, or principal topics and take a stance on one of them. Please go to the links below, and enotes can provide you with this information.

The role of the subplot would be a good thesis since critics disagree about it. Historically, critics looked at the parallels between the two, but more recently, they have emphasized their differences. So, you should be able to prove one or the other, depending on what you believe.

Enotes also has "Suggested Essay Topics" that I've referenced below. These are based on specific acts and scenes, but you might find a general one you could use as a thesis for the whole play. They also have "Sample Essay Outlines" based on the whole play that I have also referred to. They give you topics and a sample outline to help you get started. I wouldn't advise copying from them, but they can sure give you ideas about what you might include.

Some topics I would recommend are:
1. Love as it is shown in the play
2. Trace Lear's madness in the play -- when it begins, the specific events, etc.
3. Animal imagery in the play

Again, as a teacher, I usually look to the literary elements of any piece of literature to come up with a good thesis statement. Look at the themes, conflicts, and any of the literary elements that seem to be important. You can usually find great ideas by looking at them.

Good luck with your paper, and don't hesitate to ask for more help if you need it.