In good light, your shutter speed will be something like a hundredth of a second (or less), so it's basically negligible in the limitations of continuous-drive. There's some inherent limits from moving the mirror and resetting the shutter, but the primary limits are processing time and writing the data.

It takes some time to read the sensor data and to convert it into JPEG. This is usually quite fast (in fact, in cameras in the past half-decade, it's basically a non-issue), but if you enable features like in-camera distortion correction, the overhead can be higher.

Second, writing to the memory card itself can be slow. If you're getting much below the rated framerate of your camera, I'd check to see if a faster card will help. Of course, if you exceed the camera's other limits (including limits of the data connection), that won't help. You can also try reducing the amount of data you write: JPEG is (much) smaller than RAW, so that usually helps (the writing is orders of magnitude slower than the conversion), and of course RAW+JPEG is the slowest. You might even experiment with lower-resolution JPEGs to see if that helps,