What kind of SWR meter are you using? If it's the kind that you have to switch to adjust it to full scale (sometimes with a switch marked "CAL" or "FULL SCALE") and then switch it to "SWR", then the meter is likely the problem. It's responding to the change in forward power with a change in reflected power. If you set it for a full scale reading of ten watts forward power and it measures, say, two watts of reflected power, that two watts won't, or shouldn't, change as long as the ten watts output remains constant. But if you increase the output power to 100 watts, the reflected power should increase to somewhere around 20 watts.
This type of meter HAS to be re"calibrated" every time you change power level or frequency/channel.