How can I fix my dresser drawer?

The trouble with drawer construction is that the front and particularly the sides and the back are very thin - there is not much material to hold a screw securely. Have you considered wooden blocks and glue inside the affected corners? You will need to clamp firmly together while the glue sets - White PVA is probably best on wood but not on plastic.

If you must use metal brackets stretcher plates may work for you - cheap - used for fitting table tops...

http://www.kitchendoorcompany.co.uk/product-images/accessories/fullsize/stretcher-plate-angle-bracket.jpg

Peccavi
January 2010