checking configuration history of Turing machine using PDA
PDAs are strictly weaker than TMs. For instance, PDAs always halt after precisely $|w|$ steps, but there are algorithms (aka TM) for questions that take more steps than that.
sorting an array for example, would take definitely more than $\Omega(|w|log(|w|))$ which is larger than $|w|$ at some point.
It cannot simulate running a machine on a word as it simply takes more time! Checking configuration history takes less time because it's length is proportional to the running time of $M$ on $w$.