When dealing with infinite dimension we have to be careful with our notation since although the sets do not coincide, the closures (with respecto to any of the topologies below) do coincide.
Denoting with $\otimes _a$ the algebraic tensor product and without the subindex the closure w.r.t. any of the weak, $\sigma$-weak, strong, $\sigma$-strong, strong*, $\sigma$-strong* topologies, it is immediate that $L(H) \otimes_a L(H) \subset L(H \otimes H) \Rightarrow L(H) \otimes L(H) \subset L(H \otimes H)$.
Now, take a $X\in L(H \otimes H)$. We will show that there exists a sequence in $L(H) \otimes_a L(H)$ weakly convergent to$X$. Consider $\{e_i \otimes e_j \}_{ij}$ an orthonormal basis of $H \otimes H$. For any $n$ define the trunked operator \begin{align*}X_n&=\sum_{i,j,i^\prime,j^\prime}^n |e_{i^\prime} \otimes e_{j^\prime} \rangle \langle e_{i^\prime} \otimes e_{j^\prime}| X |e_{i} \otimes e_{j}\rangle \langle e_{i} \otimes e_{j}|\\ &=\sum_{i,j,i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime}, Xe_{i} \otimes e_{j} \rangle |e_{i^\prime} \rangle \langle e_i | \otimes |e_{j^\prime} \rangle \langle e_j |\in L(H) \otimes_a L(H). \end{align*}
For arbitrary simple tensors $u\otimes v,z\otimes w\in H \otimes H$,
\begin{align*}\langle u \otimes v ,X_n z \otimes w \rangle& =\sum_{i,j,i^\prime,j^\prime}^n \big\langle e_{i^\prime} \otimes e_{j^\prime},X e_i \otimes e_j \big\rangle \big\langle u\otimes v, |e_{i^\prime}\rangle\langle e_i |z \otimes | e_{j^\prime} \rangle \langle e_j | w \big\rangle \\
&= \bigg \langle\sum_{i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime},u\otimes v \rangle e_{i^\prime} \otimes e_{j^\prime} , X \sum_{i,j}^n \langle e_{i} \otimes e_{j}, z\otimes w \rangle e_{i} \otimes e_{j} \bigg \rangle, \end{align*} from where it follows that $\langle u \otimes v ,X_n z \otimes w \rangle \xrightarrow[n]{\ \ \ \ \ } \langle u \otimes v ,X z \otimes w \rangle $ since $X$ is bounded.
While for general tensors $s,t\in H \otimes H$, $s=\sum_{m^\prime l^\prime} s_{m^\prime ,l^\prime}\ e_{m^\prime} \otimes e_{l^\prime}$, $t=\sum_{m,l} t_{m l}\ e_m\otimes e_l$, where $s_{m^\prime,l^\prime}$ y $t_{ml}$ are
complex numbers such that \begin{align*}\sum_{m^\prime, l^\prime} |s_{m^\prime, l^\prime}|^2<\infty,\ \ \ \ \sum_{m, l} |t_{m l}|^2<\infty,\end{align*}
\begin{align*}\langle s, X_n t \rangle &=\lim_{h^\prime,h\to\infty}\sum_{m^\prime, l^\prime, m,l}^{h^\prime,h} \overline{s}_{m^\prime l^\prime} t_{m l} \big\langle e_{m^\prime} \otimes e_{l^\prime}, X_n e_{m} \otimes e_{l} \big\rangle \\
&=\lim_{h^\prime,h\to\infty}\sum_{m^\prime, l^\prime, m,l}^{h^\prime,h} \overline{s}_{m^\prime l^\prime} t_{m l}\bigg \langle \sum_{i^\prime,j^\prime}^n \langle e_{i^\prime} \otimes e_{j^\prime}, e_{m^\prime} \otimes e_{l^\prime} \rangle e_{i^\prime} \otimes e_{j^\prime}, X \sum_{i,j}^n \langle e_i \otimes e_j, e_m \otimes e_l \rangle e_i \otimes e_j \bigg \rangle \\
&=\sum_{i,j,i^\prime,j^\prime}^n \bigg \langle \langle e_{i^\prime} \otimes e_{j^\prime},s \rangle e_{i^\prime} \otimes e_{j^\prime}, X \langle e_i\otimes e_j,t \rangle e_i\otimes e_j \bigg \rangle \xrightarrow[n]{\ \ \ \ \ } \langle s, X t \rangle, \end{align*} since $X$ is continuos. Hence $L(H\otimes H) \subset \overline{ L(H) \otimes_a L(H)}=L(H) \otimes L(H)$.
So not every operator $T$ can be written like $T_1 \otimes T_2$ , but as the limit of linear combination of simple tensors.