determine if a turing machine halts in less than n step, in less than n step
Maybe the Turing Machine Linear Speed Up Theorem will answer your doubt (https://en.wikipedia.org/wiki/Linear_speedup_theorem)

Counter-intuitively or maybe intuitively, this states that given a Turing Machine which does something in say $n$ steps there are Turing Machines which can solve the same problem in $n/k$ steps for any constant $k$ independent of $n$. These Turing Machines would basically run $k$ steps of the original TM as $1$ step, hence having more state space, but linearly faster.

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