Please help! I changed the light and it works fine, but now when it's on and I turn on the exhaust fan, the fuse blows. Everything else on that fuse works fine. What could be going on? I'm baffled by this

You may have misswired the fan. Somehow you have a hot and neutral connected through the switch. It may have been a switch loop but we don't know how it is or was wired. Tell us the wire colors on the fan switch and how they are connected to the fan.

Thanks for the quick feedback. I didn't change anything on the fan, only changed the bathroom light which is on the same circuit as the fan. The light works fine, as long as the fan is not turned on. I turn the light on, all is good. Turn on the exhaust fan and a little pop noise and the fuse blows and I got to go down to the fuse box to reset the circuit

I can only assume that somehow you misconnected a wire on the light circuit so that the hot connects to ground or neutral , bypassing the motor for the fan. If there are 2 switches, there should be 2 hots, one for the fan and one for the light and a common or separate neutral. Unfortunately, we cannot see the wiring. Obviously it is in the fan side or it would have tripped the breaker (you say fuse but assume breaker or a resettable fuse). Again, how it the switch for the fan wired, 2 blacks or a black and a white-possibly with black tape on it?

Okay, obviously we need some additional information here.

Are the light and fan two separate devices controlled by two separate switches or is it one unit?

The previous suggestions are really spot on. If all things in this equation (fan and light) were working properly originally and all you did was swap out the existing light for a new unit, then more than likely neutral and hot were cross connected within the circuit extension feeding the fan.

Again, because switch loops, note most switches do not use a Neutral connection. They simply interrupt the hot connection and route to the switch and in turn to the device. (white going to the switch is connected to the black hot feed conductor and if that supply feeds another device, there would be two black and one white. The white in this case is "re-tasked" and must be marked with black tape or marker to show that it is being used as a hot conductor.) (black coming from the switch is connected to the light's black)

The symptom you are describing is that of a neutral short which is indicative of a miss-wired switch unless the fan unit itself is now defective.

They are controlled by separate switches in a double switch box. Inside it's a sea of wires, with 3 white in a connector, 4 black in a connector and at least 1 white and another larger dark wire with a cloth looking exterior in a third connector. Each switch has 2 blacks connected to it. Could the light still work if reverse-wired? Both wires were black but I was sure I knew which was hot after removing the old light. I assumed it wouldn't work if wired in reverse, so I didn't check that

There is no "reverse" in a switch. You can have power on the top or bottom. You mentioned a white and "dark" wire together. Where do they go? Cloth exterior would indicate a fairly old house. If you have only black on the side by side switches, it would indicate 2 separate power feeds, rather than a shared neutral. You need a test meter to start checking where power comes from and where it goes.