I say to my friend Mark: "You should read this book". Later, Mark talks with another person about this and says: "This book was suggested to me by a friend".
Is this correct? Does it sound natural or is there another way to put it?
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You'd better to use recommend instead of suggest, because there is a slight difference between their meaning [1 and 2] and Google books results confirms this claim. (961 results for "recommend me a book" and 322 ones from "suggest me a book")
"We use “suggest” when talking about giving someone an idea in general, and we use “recommend” when telling someone that a certain choice or option is the best one". 
And this is worth mentioning from YAHOO! answers:
" if I make a recommendation, that implies that I have experience with the subject at hand, and have handled it with positive results. It also implies that if its good enough for me, then it should work out for you".
In this instance, recommended is a better option than "suggested".
If you want to use "suggested" you could say:
My friend suggested that I read this book.
Yes, suggested fits in to this context. You could also say recommended
"This book was suggested to my by a friend." Is perfectly fine. An alternative would be to say "A friend suggested this book to me." The second sentence puts more emphasis on the friend, whereas the first sentence focuses more on the book. A lot of this is active/passive voice and I learned about from here http://writingcenter.unc.edu/handouts/passive-voice/. Whichever sentence works better for what the speaker, writer is trying to communicate is usually more effective. However, some sticklers English teachers in the U.S. absolutely do not like passive voice.
Cred: Native speaker, B.S. Business,