A question on the linear speedup theorem
I didn't check your working but your conclusion isn't at all absurd.  For every Turing machine that decides&nbsp;$L$, and for every $x\in\Sigma^*$, there is another Turing machine&nbsp;$M_x$ that decides&nbsp;$L$ and that runs for exactly $|x|+1$ steps on input&nbsp;$x$. Note that this is more than twice as fast as your conclusion of $2|x|+4$. This also means that, for all languages&nbsp;$L$ and all strings&nbsp;$x\in\Sigma^*$, $t^L(x)=|x|+1$.

$M_x$ operates as follows. First, it reads the first $|x|+1$ characters of the input.  If these are the string&nbsp;$x$ followed by a blank, it accepts if $x\in L$ and rejects if $x\notin L$; if they are not $x$ plus a blank, it returns to the left-hand end of the tape and does whatever $M$&nbsp;would do. Note that $M_x$&nbsp;does not need to contain any algorithm to decide whether $x\in L$. If $x\in L$, the machine is programmed "If my input is $x$, accept it; otherwise, rewind and do what $M$&nbsp;would do.", and if $x\notin L$, the machine is programmed "If my input is $x$, reject it; otherwise, rewind and do what $M$&nbsp;would do."