Hi John - Your other site is a self-hosted WordPress site, which is different from sites hosted here at WordPress.com. You likely use a plugin or theme-specific setting of some kind to plug your images into galleries on that site.
The issue with your image-heavy page is that you have things like the following: Here are three images, and they are right- left- and right-justified respectively:
If you had a lot of text in there, it would wrap around them, but you don't, so the images are just sort of wrapping around each other and they look odd.
You could fix this by: (1) left-justifying all three so that they sit in a line, but as I said, you'd have to monkey with the sizing, so that they didn't drop down below each other; (2) centering them, so that they display on top of each other in a vertical line; or (3) putting them into a gallery.
Here's how they look when I put them in a gallery (I don't think I got the same three images, but this is just as an example):
There are also many other gallery styles you can choose if you don't like that one.
Can you please go through that post and try reformatting some of your images as suggested? Here are instructions on how to set up a gallery:
And here's information about how to make images sit beside each other on a single line:
If you get stuck, let me know. I'm happy to work with you until you get your images appearing correctly.