  The Tur\'{a}n problem asks for the largest number of edges ex$(n,H)$ in an
$n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of
the most important problems in extremal graph theory. However the order of
magnitude of ex$(n,H)$ for bipartite graphs is known only in a handful of
cases. In particular, giving explicit constructions of extremal graphs is very
challenging in this field. In this paper, we develop a polynomail resultant
approach to algebraic construction of explicit extremal graphs, which can
efficiently decide whether a specified structure exists. A key insight in our
approach is the multipolynomial resultant, which is a fundamental tool of
computational algebraic geometry. Our main results include the matched lowers
bounds for Tur\'{a}n number of $1$-subdivision of $K_{3,t_{1}}$ and linear
Tur\'{a}n number of Berge theta hyerpgraph $\Theta_{3,t_{2}}^{B}$ with
$t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the random
algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph
theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the
known estimation for the smallest value of $t_{1}$ concerning a problem posed
by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions,
Combinatorica, to appear], while the constant $t_{2}$ improves a result of He
and Tait~[Hypergraphs with few berge paths of fixed length between vertices,
SIAM J. Discrete Math., 33(3), 1472-1481].
