Whenever I draw on a larger canvas, like 1800x2400, all the brushes lag. Is there an easier way to fix this or would I have to resort to drawing on a much smaller canvas size with smaller brush size?

Well, in the first place: that is not a large canvas. If you experience brush lag with a canvas of that size, something is wrong with your system. You didn't give any information, so I'm just guessing here. Check whether opengl is enabled, and whether Krita feels faster if you enable or disable it; some graphics drivers are just plain ****. Check whether something is running in the background, some virus scanners take a lot of CPU for instance. Check whether playing with the precision parameter in the brush settings makes a difference. Check the memory item in the status bar whether you're already swapping a lot (shouldn't happen, not even on a 32 bits Windows with images this small).