There an old joke that goes like this:
A fellow goes to New York to attend a concert, but gets lost. He spots someone carrying a violin case.
“Excuse me, Sir, can you tell me how to get to Carnegie Hall?”
The musician smiles and says, “Practice, practice, practice.”
Congratulations on getting this far with learning to read Thai. However, this is only the beginning of your journey. By this stage you should be able (slowly, very slowly) to decrypt simple sentences. What now is needed is practice, practice, practice.
If you're living in Thailand, try and read all the Thai that's around you: shops signs, street names, advertisements, &c..
Read simple texts prepared for foreign learners of Thai. (These are to be preferred over children's books which have a much wider vocabulary, and so will have you looking words up in a dictionary time and time again.) Two readers are:
(1) Thai Reader by Mary R. Haas, ISBN 9780879502645. (The cost of a new copy is ludicrous c. US$800, but second hand copies are much, much cheaper.) This reader is also available online (and for free) on this site at https://thai-notes.com/reading/thaireader.html and at the SEALang Site at http://www.seasite.niu.edu/thai/thaireader/frameset.htm.
The SEALang Site also has the reader with parallel English translation at http://www.sealang.net/lab/justread.
(2) The University of Wisconsin developed texts primarily to be used in a classroom setting. However, they can be useful to the independent learner. >http://www.readingthai.wisc.edu/thai-reader-site-home.html.
Other simple texts include folk stories. The following site includes Aesop's Fables, English fairytales, Thai folktales, and Jataka stories (i.e. accounts of the lives of previous incarnations of the Lord Buddha) https://www.nithan.in.th.
Millions of Thai children have learnt to read at school from the much loved Manee series of 12 textbooks (though they have now been replaced in schools). They are available online from many sources (just search for "manee books"). One place where you can download the books as .pdf files is Dropbox.
Undoubtedly, at the early stages you'll need frequent recourse to a dictionary. I'd suggest you read how to do this correctly from my notes at https://thai-notes.com/notes/dictionaryuse.html. (If you don't already know the alphabetic order of Thai consonants you'll probably benefit from reading my notes at https://thai-notes.com/notes/consonantorder.html first.)
Reading Thai handwriting represents a major challenge. Even though schoolchildren are taught to write beatifully, when they grow up their handwriting is often little more than an incomprehensible scrawl. J. Marvin Brown wrote about and provides copious examples of handwriting in his book Thai Reading (Text Mostly Reading), published by AUA Language Center.
These days we rarely need to handwrite in Thai. We do, however, need to be able to type it. For that I'd suggest my Typing Trainer which has graded 52 lesson. It's at https://thai-notes.com/typing/typingtrainer.html.
Good luck!