We prove that relative to a random oracle answering $O(\log n)$-bit queries, there exists a function computable in $O(n)$ time by a random-access machine (RAM) but requiring $n^2/polylog(n)$ time by any multitape Turing machine. This provides strong evidence that simulating RAMs on multitape Turing machines inherently incurs a nearly quadratic time overhead, since the random oracle can be heuristically instantiated by a concrete hash function such as SHA-256.