How to conceive a Turing machine that is the intersection of the languages of two Turing machines?
What you ask might not be possible in general. If $L(T)$ denotes the set of words $x$ for which $T(x)$ accepts, and you do not have the additional assumption that $M$ and $M'$ recognize $L(M)$ and $L(M')$, respectively, then you could choose:

$M$ as a Turing machine that takes another Turing machine $T$ as input and accepts if and only if $T$ halts on empty input. (Notice that such a Turing Machine $M$ exists).
$M'$ as a Turing Machines that recognizes $\Sigma^*$.

Then, there is no Turing Machine $T$ that recognizes $L(M) \cap L(M') = L(M)$ since $T$ would solve the halting problem.
On the other hand, if you are fine with a Turing machine that accepts $L(M) \cap L(M')$ or if you have the additional assumption that $M$ and $M'$ recognize $L(M)$ and $L(M')$, respectively, your problem is solved by a Turing machine that, on input $x$, operates as follows:

Simulate $M(x)$ until it halts (possibly never).
If $M(x)$ rejects, reject.
Simulate $M'(x)$ until it halts (possibly never).
If $M'(x)$ rejects, reject.
Accept.