In his 2011 paper Peter Keevash surveys the known results and methods, and discusses some open problems: Hypergraph Turán problems, Surveys in combinatorics 392 (2011): 83-140.
In particular, you might want to look at flag algebras. See, for instance
Falgas-Ravry, V., & Vaughan, E. R. (2013). Applications of the semi-definite method to the Turán density problem for 3-graphs. Combinatorics, Probability and Computing, 22(01), 21-54, doi:10.1017/S0963548312000508
Razborov, A. A. (2013). Flag algebras: an interim report. In The Mathematics of Paul Erdős II (pp. 207-232).