The State Library of New South Wales has many indexes and databases that can be searched from home if you are a New South Wales resident and have a State Library Card. One of these is the 17th-18th Century Burney Collection Newspapers (Gale). As the title suggests it has titles that were published around 1603 to the early 1800s.
They are predominantly London newspapers with some from the British Isles, and colonies. There are over a million digitised pages. Newspaper images can be magnified and you can save and print the articles. There is also a help page that shows you how to search, print and email articles. The “About” link tells you a history of the Reverend Charles Burney (1757-1817) who collected these items and about the collection.