I had a terrible time with that until I bit the bullet and bought a Golden Feather Membership. It's not expensive, it's available in the BYC store, and with it you can turn off the ads completely!! Best thing I ever did on BYC.
As I see it there are two drawbacks to the GFM....you have to use PayPal to pay for anything you buy on BYC, including the membership, and PayPal hates me! And there's no way to know for sure without asking when your GFM will expire, unless you have the memory of an elephant or there's some secret to it that I haven't figured out yet.
So that's what I'd do. This is a big site, with lots of members and lots of stuff going on. The folks who work hard to keep this site clean and civil (not that it takes much with most of the folks on here, they behave pretty well) are all volunteers from what I understand. Running a web site this big needs some kind of financial boost, which comes in part from those ads. A GF membership is a pretty small price to pay for all we get!