I would disagree with this physician. The most common cause of this would be a urethritis- caused by a sexually transmitted organism. You need to get swabed and tested for this. It is still possible that a urinary tract infection can cause this, but that would be less common seeing that there is a purulent discharge from your penis. You would have to have A LOT of pus (and therefore be very sick) to have this happen.
I would not start the antibiotic until seeing your doctor and seeing what the cause of this is. Taking it now may cause some partial treatment and mask the cause..
You are not putting yourself at risk by waiting one more day. It has nothing to do with her finishing up her period. I would definitely wait and get swabbed to make sure what this is... If you can go to an urgent care clinic or even the ER, they can do it there.
Of course, if you have any back pain or fever or chills, you need to be seen sooner.
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Listen- I am an ER doctor and I know the facts. It is possible that she may have slept with someone else. You have the signs and symptoms of someone with a sexually transmitted disease. Yes, it is still possible that this is a urinary tract infection, but if you take amoxil for a STD, you will not be treating it properly. That being said, there are also nonSTD causes for urethritis- such as a virus or even autoimmune. But you need to be swabbed, period.