About 6 weeks ago my traffic stats jumped tremendously. I started noticing "top searches" for variations of porn words. In looking at my top landing pages in Google Analytics, it is showing landing page URLs ending in snippets such as "-cup-girl-porn-gif-xxx_e3ba94.html". There are over 1000 of them! Now Google webmaster is trying to crawl them and, of course are getting 404 errors. I have asked my hosting to check for malware on my server and they can't determine the cause. The increase in traffic is close to maxing out my server capacity (requiring me to upgrade). I was normally seeing daily traffic of around 150-200, but since this issue began, it is nearing 2000 daily. How do I get rid of these fake (and pornographic) URLs?
Based on what you have put in your question it would appear most likely that your site has been infected with malware. There is a broad range of malware out there which will only show content to a restricted range of requests but not to others or to search engines. This most often happens from insecure file uploading to your site but this is by no means the only way that this happens. Best thing to do at this point is to shutdown the site, erase everything, and reinstall from backups. This will give you a hopefully clean implementation (if you have a backup of the site before infection), and beyond that it would be strongly recommended to do a security health check of your site and secure any infection vectors including file upload. You should also implement antivirus and malware scanning on your webserver with regular scans and a realtime scan of all uploaded files prior to saving them on your server to aid in protecting your site from future attacks.