The noise properties of single-walled carbon nanotube transistors (SWNT-FETs) are essential for the performance of electronic circuits and sensors. Here, we investigate the mechanism responsible for the low-frequency noise in liquid-gated SWNT-FETs and its scaling with the length of the nanotube channel down to the nanometer scale. We show that the gate dependence of the noise amplitude provides strong evidence for a recently proposed charge-noise model. We find that the power of the charge noise scales as the inverse of the channel length of the SWNT-FET. Our measurements also show that surprisingly the ionic strength of the surrounding electrolyte has a minimal effect on the noise magnitude in SWNT-FETs.