a) I checked a few of your images, and they all seem to be in a 4/3 or 3/4 aspect ratio. But some of them are landscape-shaped and some are portrait-shaped. I don't know if you wanted/tried to put a landscape-shaped and portrait-shaped image side by side: this would never look good, in my opinion, but anyway if you did that and the two inserted versions didn't have the same height, then next image would show up below the shorter image instead of in a new row (because of the wrap-around effect). To prevent this from happening, and make sure the images would show in rows of two, you'd have to switch the editor to Text (=code) and paste this after each pair of image codes:
(edited to fix the code: by Tess)
b) Two images side by side would look ok if they were both landscape-shaped or both portrait-shaped. In the theme you're using, the maximum width that will allow two captioned images in a row is 269 pixels. So you'd make things easier if you went to Settings > Media and set the default width for the medium option to 269; then you'd simply select that option when inserting such images, and pairs of left/right aligned images would show side by side (provided you inserted them with no space or line break or paragraph break between them).
c) But actually both #a and #b are unnecessary. Since you insert several images with no text after each image, just use the gallery feature instead of inserting individual images, selecting a two-column display.
(Again you may need to go to Settings > Media and adjust the default widths.)
d) And a more important tip: the files you upload are unnecessarily large. The ones I checked are 3264x2448 or vv (while the maximum that can fit in the theme you're using is 610 pixels). This means you waste your storage space, you make your pages load slower, and you get slightly reduced quality on the post. Ideally you should use an image editing application to downsize copies of your images to the desired width, and upload and insert those copies instead of the oversized originals.