For a positive integer 𝑡, let 𝐹 𝑡 denote the graph of the 𝑡 × 𝑡 grid. Motivated by a 50-year-old conjecture of Erdős about Turán numbers of 𝑟-degenerate graphs, we prove that there exists a constant 𝐶 = 𝐶(𝑡) such that ex(𝑛, 𝐹 𝑡 ) ⩽ 𝐶𝑛 3∕2 . This bound is tight up to the value of 𝐶. One of the interesting ingredients of our proof is a novel way of using the tensor power trick.