Pics are over saturated on mobile devices

When you say "every large screen" it sends the message you do not understand colour management, that you have not calibrated your monitors, that you have just been adjusting colours to look right on monitors rather than using colour management. Is this the right message?

There is only one answer to this, and it's a very short one:

This is why color management was invented.

If the device/application doesn't do color management, that's that.

I think there isn't a real universal solution to the problem, because the color final look depends on too many variables: embedded profile, color management engine of the device, choosen profile on the devices, correct calibration of the device, etc...

Pictures look oversaturated on phone: Beginners Questions Forum: Digital Photography Review

Please take a look at this very interesting thread:

Images look oversaturated on other devices after export