Continuing the road analogy a bit. The Internet connection is the expressway that can accommodate lots of cars traveling between cities at a reasonable speed.
Regardless, the customer's throughput to a given site will also be determined by the site's performance. As I update packages, the download time for a given file size can vary from a few seconds to several minutes -- depending on the remote server's current load and the intermediate node loads.
Also a gigga-speed connection may not result in websites loading much faster because a page may include connections to text and images from multiple servers, each requiring some time to connect and respond. For a while I had an older computer on a gigabit connection, but webpages were pokey because the poor old machine was overwhelmed by the number of scripts, images, and connections. The new computer could display the page in a blink.