We show that for all functions \(t(n) \geq n\), every multitape Turing machine running in time \(t\) can be simulated using only \(O(\sqrt{t \log t})\) space. This is a substantial improvement over Hopcroft, Paul, and Valiant’s simulation of time \(t\) in \(O({t}/{\log t})\) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size \(s\) can be evaluated on any input in only \(\sqrt{s} \cdot poly(\log s)\) space, and that there are explicit problems solvable in \(O(n)\) space which require at least \(n^2/poly(\log n)\) time on every multitape Turing machine, thereby making a little progress on the P versus PSPACE problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].