My two main suggestions are to make smaller improvements to the article and to use more than one source. Unless you are very skilled at proper paraphrasing, using a single source greatly increases the risk of close paraphrasing, i.e. improper paraphrasing, and also leads to a very one-sided view of the subject. I've added two reference to the article with links to their pages on Google books (most of which are visible, at least in the UK). The print copies may also be in the Longy library:
Marco, Tomás (1993). Spanish Music in the Twentieth Century. 
Draayer, Suzanne Rhodes (2009). Art Song Composers of Spain: An Encyclopedia
The previous reference I had added—Romero, Justo (2004). Liner Notes: Joaquín Turina (1882–1949): Piano Music, Vol. 1—is also very useful, especially for discussing his musical style and works chronology. I suggest you use these as your sources for further expansion with the biography on joaquinturina.com] as a back-up only.
 
One thing you could do is separate out the last paragraph (beginning with "His works include...") with its own heading Music and expand the commentary there. For example, he was a very noted composer of art song, which is given short shrift in the article. Much has been written about his approach to incorporating Spanish musical idiom in his work and on his style in general, including its strengths and weaknesses. This is also lacking in the article.
 
Another possibility, is to separate out a section at the beginning of biography to Early years where you discuss his family background and his musical education prior to Paris. The material between that section and Music could be entitled Paris and Madrid and can always be expanded in the future.
 
A third possibility is to expand and rationalize the list of works in the article. The current one has three major problems. First, it excludes many of his important works because it relies solely on those scores available at IMSLP. Secondly, it gives no indication of each work's type. Thirdly, it gives no indication of the date of their composition. Listing them alphabetically, is not a good way to go. If you do decide to concentrate on improving the works list, I suggest using List of compositions by Claude Debussy by genre as a model, although in this case, the list would not be a separate article, but rather continue to appear in the Selected works section of Joaquín Turina.
 
Hope that helps. Best,