Images / Directory Problem, Absoulte URL vs Relative URL

I am having an issue with using images located in my images directory inside of my theme folder

This blog here works properly: https://buenolisto.com/

When I code for an image I only need the directory structure that looks like this: images/bg_header.jpg

Even though the full url is: https://buenolisto.com/wp-content/themes/bueno-listo/images/bg_header.jpg

The problem I am having is on this blog: https://www.foreclosuresnewportrichey.com/

On this blog I cannot get an image to display from the image folder within my theme: https://www.foreclosurenewportrichey.com/wp-content/themes/twentyten/images/big_tim_circle.jpg

This links give me an invalid request message

Ideally, I want to code my site with a directory structure similar to the bueno-listo site shown above whereby I only need this in the code to render the image:

images/big_tim_circle.jpg

Right now I have all the images coded in for absolute URLS from the media folder and that is not easy or efficient.

I know there must be some settings I am missing some where, any help would be appreciated.