You may be leaning too far forward.
Shoulders up, elbows up and chest out.
Make sure you rack the bar close enough to your throat and not on the "meat" of your shoulders where the bar is more likely to slide forward.
A note on the clean grip - you really only need the tips of your pointer and middle fingers under the bar. The fingers are not there to hold to bar in place, the are under the bar to act as a fulcrum so you can create leverage to hold your shoulders and elbows up high and tight.
Be very aware of your elbow height. People tend to drop them a lot when they reach the bottom position of the front squat.
My advice, become proficient with the clean grip. It is the best way to front squat.
Check out this video to develop your wrist flexibility.