"Read" is one of those words which don't change between present and past when written, but you probably heard that the present form has a long vowel, yet the past form has a short vowel. In this case, you also have the infinitive: "to read" (long vowel). "I read book" is broken English. This can't work at all. You either need "I read a book" (which is past tense, unless you mean it as a habit: "when I am bored, I read a book") or "I read books" (books in general, which tells me this is present tense). This is not as difficult as it seems. ;)

She started to read (this cannot be past, to+V present) her book. It was a good book. She read(past tense) one page. Then she read(past tense) another page."

There is nothing wrong with the text. It is correct. She started to read her book. Here ''start'' is the main verb of the sentence not ''read''. So you cannot use the past form of ''read''.It should be present in this sentence.