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thumb through (something)
To casually peruse or look at the pages in a book, magazine, or other reading material without much attention or scrutiny. I thumbed through the sports section while waiting at the doctor's office, but I was too nervous to pay attention to it. I thumbed through your proposal, and even at first glance I can tell you that it won't be approved by the board.
thumb through somethingand leaf through something
to look through a book, magazine, or newspaper, without reading it carefully. I've only thumbed through this book, but it looks very interesting. I leafed through a magazine while waiting to see the doctor.
To go through some reading material quickly or superficially, turning from page to page with or as if with the thumb: I thumbed through the directory for my dentist's phone number.