Wiring for light switch and exhaust fan
The problem may well be the "push ins", aka "backstabs".

Ignoring grounds (green or bare, all go together), you have:


Hot to switch 1
Jumper hot from switch 1 to 2
Switched hot 1 (light)
Switched hot 2 (fan)


Rewire as follows:


Take hot to switch 1 and connect two short pieces of black (or any color but white, gray, green or bare) wire and connect them with a wire nut.
One of the short pieces goes to the "hot" screw terminal on switch 1
One of the short pieces goes to the "hot" screw terminal on switch 2
Switched hots are connected as before, but make sure to use screw terminals.


And if you are putting in new smart switches/dimmers (don't use a dimmer on a fan), motion sensors or anything else "special" that needs a neutral, you can add a short white wire to the bundle of whites in the wire nut. Replace with a larger wire nut if necessary. The other end goes to the "neutral" screw terminal on the switch.