A book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. The book thickness $bt(G)$ of a graph $G$ is the smallest possible number of half-planes for any book embedding of the graph (also known as pagenumber and stacknumber).
A brief introduction without proof in the Literature [1] (page 60) tells us that $bt(K_{2,2,2,2})=4$. It also gives its a four-page book embedding of the $K_{2,2,2,2}$ as below.
Of course, its 4-page four-page book embedding is not unique.
[1] S.G. Kobourov, G. Liotta , F. Montecchiani, An annotated bibliography on 1-planarity, Computer Science Review, 2017, 25: 49-67.
According to the following theorem, we can know $bt(K_{2,2,2,2})\ge 3$ (note that $K_{2,2,2,2}$ is a non-planar graph). Combing this with the above 4-page embedding, we have $ 3 \le bt(K_{2,2,2,2})\le 4$.
THEOREM 2.5. [2] Let $G$ be a connected graph. Then
(i) $bt(G) = 0$ if and only if $G$ is a path;
(ii) $bt(G) \le 1$ if and only if $G$ is outerplanar;
(iii) $bt(G) \le2$ if and only if $G$ is a subgraph of a hamiltonian planar graph.
[2] F. Bernhart, P. C. Kainen, The book thickness of a graph, Journal of Combinatorial Theory, Series B, 1979, 27(3): 320-331.
But how do we prove that $K_{2,2,2,2}$ does not have a three-page book embedding? So far, I haven't seen a theoretical proof.
A genenal question to ask: Has the book thickness been determined for multipartite graphs?
Theorem 3.3 in Bernhart and Kainen's paper defining book thickness (that is, in citation [2] in the question) states that the book thickness of a graph with $p$ vertices and $q$ edges is at least $\lceil \frac{q-p}{p-3}\rceil$, which gives the lower bound $\lceil \frac{24-8}{8-3}\rceil = \lceil 3.2\rceil = 4$ in the case of $K_{2,2,2,2}$.