Quality over quantity
Please note that I am not an expert in Japanese university exams and my answer here for you is based on my experience as a graduate student in the US. As you probably know, those graduate programs are very demanding and we are required to read tons of books etc. which are definitely not ‘non-native friendly.’ Please also note that I am not at all a bookworm type of person, so my advice can be quite different from other experts’.
To improve your reading comprehension, I would recommend you try to write as much as possible. Think about improving your listening comprehension - it is useful to check it by your speaking/oral performance, and the same goes for reading and writing. Judging from your post, you seem to have pretty good writing skills, so I think writing exercises I describe below would work well with you.
Start with summarizing what you read. As you read, I suggest you underline the parts you find important or interesting. Your first draft may be lengthy, but that is ok. Revise your draft to make it short and sweet. After writing up a good summary, the next step will be writing an essay. While reading, take notes on your random thoughts in English. When you organize your thoughts, it may be helpful to copy the underlined parts with your notes in a separate sheet and sort them out.
You may be seeking advice on something more to read, but I would say quality matters more than quantity. If you still have the magazines you listed, pull them out from your bookshelf and select one (yes, only one!) article you find most interesting. Read carefully, summarize, and write an essay on it. Ask someone to edit your writing. Whenever possible, have a discussion with your friends on what you read and write. Re-read the article and you will be amazed at how well you comprehend it. I would call it ‘in-depth reading’ and believe it helps improve your ability.
Hope this helps!