Spray the area with bleach/water or other and see if it takes care of it before replacing the sheetrock. If the cleaner looks like it worked, wait a little while and see if it returns. If it returns, then remove sheetrock. It could be mold/mildew on the surface or it could have penetrated deep into the sheetrock.

You could also try a small dehumidifier.

https://learn.allergyandair.com/bathroom-mold/

https://www.amazon.com/customerpicks/Explore-room-dehumidifiers-for-bathrooms/aaac062b53991f6f0a76?ref_=ws_cp_a4c140d12e5e593c9210_cp_3_t

Concrobium is the preferred product.

Hi Flo, this is on the second floor and no bathroom above, just attic. And these leaks are on an exterior wall. Do you think that it could be exterior water coming in?

Is there a "stack" or roof penetration? There might be water coming in around that. Maybe a vent for a hot water heater?? Even Tankless water heater need vents. How is condition of roofing? You have to identify the source of the water and fix that, then you can do topical removal (with proper protection, etc.). I have used Tea Tree oil successfully to remove and prevent. BUT, it has strong odor (non-toxic, just darn strong!). There are other treatments, but you don't want the spores being spread around or as little as possible.

Do you have an exhaust fan in there? if not get one and run it every time the shower runs.
Looks like mold....but I'll bet a coat of Killz will fix it so you can [paint over.

Are you in an area of the country that has cold winters? This would explain the problem being worse on the outside wall when there is no vent to exhaust the moisture created from taking showers. More moisture condensing on the cold wall in the winter. It could also be a leak on the outside wall. Are you having a problem with ice dams above the wall here?

If you have problems in the bedroom then you need to find out why you have moisture problems there. A leak into the wall, too much moisture in the house (are you running a humidifier in your bedroom? Perhaps you need to set it lower or run it less. Mold always has something to do with excess moisture.