I am willing to do almost anything to solve this for my family as we have been dealing with it for months.
To date I have replaced the modem and the router. The issue didn't stop after switching both.
My ISP is COX. Previous router was an Asus AC68U. It worked well at 1st but after the pron stuff started popping up, I tried switching on the filters in the router to block the porn. It didn't stop the image search pop-ups. I had reset the router a number of times hoping to stop the issue.
Purchased a new Netgear cox compatible modem in May 2017. Also purchased a new Pepwave Surf Soho router. It is a commercial grade router with supposedly better software and security. The firmware was just updated 2 weeks ago. I found out today that the firewall wasn't configured correctly. That was corrected today.
I can reset the router tonight. Usually when I do this, I will also change the SSID.
Also, you should know that we get similar pop-ups on our smart phones even with wifi turned off coming in through the phone carriers data connection. This is were it gets more strange, it only occurs when we are near our house.