Are every problems in EXP karp reducible to any EXP-Complete?
Are every problems in EXP karp reducible to any EXP-Complete?


Yes. That's the definition of $\mathrm{EXP}$-completeness.


  My question is: where is my mistake?


Your mistake is in believing that $L''\notin\mathrm{EXP}$. You've demonstrated a Turing machine that decides $L$ in time $O(n^{\log n}) = O(2^{(\log n)^2})\subset O(2^n)$. I'm not sure what you mean by "$M$ prevents the reductions"&nbsp;&ndash; you're trying to reduce from $L(M)$ to&nbsp;$L$, not from $L$ to&nbsp;$L(M)$.

On the other hand, $L''$&nbsp;is not $\mathrm{EXP}$-complete, by the time hierarchy theorem. Any language reducible to&nbsp;$L''$ must be in time $O(2^{(\log (n^c))^2})$ for some&nbsp;$c$, but this is just time $O(2^{(c\log n)^2})=O(2^{(\log n)^2})$.