Images show up once, then have to clear styles folder to see again? So I'm trying to get a site setup with a template (global news), which requires Drupal 7.x, so I'm using 7.41. Anyway, I have all the modules set up and the general layout as it should be. However, I'm having a problem with getting the images to render. 

It seems as though if I delete the /default/files/styles folder, the images will generate and show up on the website, however if I refresh the page the images fail to load until I delete the folder again. If I don't delete the folder I get a 403-forbidden for each image I'm trying to load.

It's weird, and I've almost driven myself crazy trying to get this figured out.

Any advice?
The fact that derived images can be fetched when they are absent from the filesystem but not when they are present indicates this is a webserver configuration issue.

The webserver rules are normally be set up so that any non-existent paths are routed to Drupal. When image thumbnails are generated they get served by Drupal, but once on disk they should be served directly from the filesystem. Check your .htaccess files (if using Apache) and somewhere there is likely to be an rule denying access to that area of the filesystem.