Several verbs are applied to reading material on websites like Twitter.
Reading pertains generally to text or to the contents of something published. Among Merriam-Webster's many definitions of read is a sense that the medium of reading might vary: letters, symbols, printed words, books, and situations are all mentioned as entities that can be read.
Read is widely applied to multimodal formats like websites and social media posts. People who study literacy, rhetoric, and composition regularly apply the term reading to describe what we do with websites (example). Search results show the usage "read Twitter" is widespread. As a native user of English who uses multimodal texts in his teaching, I readily associate Twitter with reading.
Other verbs might also work to catch a specific aspect of reading, like perusing or scanning. If the behavior you're looking for is reading information online, then browsing would apply. Browsing Twitter would be the same as browsing any other website. It could suggest a more superficial, skimming form of reading (see def. 2), as happens with screen reading and clicking through hypertext. The verb is widely applied to looking through internet sites (see def. 3). It is related to a name for site-display programs like Chrome and Firefox: web browser.
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