A tight cycle is an r-uniform hypergraph analogue of a cycle. Formally, the length-k tight cycle C^(r)_k has k>r vertices v_1 ... v_k and edges v_{i+1} ... v_{i+r} for each residue i mod k. We show that for all k sufficiently large and not divisible by 4, the Turán density of the tight cycle C^(4)_k is exactly 1/2. That is, we show that the densest 4-uniform hypergraph on n vertices with no copy of C^(4)_k has edge density 1/2 + o(1) as n → ∞. A key ingredient in the proof is a hypergraph analogue of the result that a graph contains no odd cycles if and only if it is bipartite, which we hope will be helpful towards understanding the Turán densities of other cycle-like hypergraphs.