Bath room exhaust fan not working. The light is working on a separate switch right next to the fan switch

You can use the switch to operate both the fan and light, if you so choose. In fact, this is a common feature in single user public restrooms. As long as the switch and wiring are rated for the amount of current, which unless you get a huge fan, it should be.

If you have the ability to install a new cable (or pull an extra wire, if the wires happen to be in conduit), you could use a device that has two switches that fit in a single gang box. Which would allow you to control each device separately.

Without actually being there, and not being able to see the wiring at each of the devices, it's difficult to be sure what's going on. But here's my best guess, based on past experience.

To make it easier, I've numbered the ungrounded (hot) conductors in the photo from left to right.

1. You've already established goes to a light.
2. Supplies power to the switches.
3. Likely goes to the fan.
4. Also goes to the fan, but since the fan doesn't have a light built in, it's likely capped off at the fan.
5. Probably feeds some other device, maybe a receptacle.

Why 3 and 5 are controlled by the same switch, I have no idea. Again, I can't see the room, so I have no idea what equipment you have.

If 1 and 4 are both for lights, it makes sense that a singe switch controls them.

One other thing to check, is to make sure the terminals of those switches are rated to accept more than one conductor. There should be a label on the switch that tells you what type, and how many conductors can connect.