Renewing Residence Permit (how?) (Istanbul)

I've read the other postings about renewing visas but haven't seen a specific response to the following:

1. My permit says to renew "within 15 days FOLLOWING the expiration." Bad translation or do I have a 15-day window to renew?

2. The online appointment generator for NEW permits specifically says to make an appointment with my district police for my renewal (i.e., not with the people from whom I originally got the permit). And herein lies my problem -- HOW do I make an appointment, or is renewal a walk-in-and-wait event?

If you've done a renewal yourself, I'd appreciate information on your experience. Unfortunately for me everyone I know in Istanbul with a permit has an employer who does the legwork for them.

I'm a US citizen living in Besiktas district.

Thanks in advance for your help.