Besides car info, you don't describe conditions when you see gauge go from M > L - FANS ON - L > M. Usually fans activate on low speed with a one letter movement on gauge when car is idling.
If temp is on M for normal driving then everything sounds normal, but a check of slow speed fan operation, as previuosly mentioned, seems prudent. Finger tips on both high and low speed relays (engine compartment fuse box) as car heats up and fans come on should tell the story. Slow speed utilizes a dropping resistor mounted on lower driver's corner of fan shroud which has been a problem for some members.
Believe early models had recall re fans/resistor best to check VIN at dealer to determine if applicable to your car.
EDIT: Please respond on member's duplicate thread responses continue here