For a positive integer $$\ell \ge 3$$, the $$C:\ell $$ -Contractibility problem takes as input an undirected simple graph G and determines whether G can be transformed into a graph isomorphic to $$C:\ell $$ (the induced cycle on $$\ell $$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $$C:4$$ -Contractibility is $$\textsf{NP}$$ -complete in general graphs. It is easy to verify that that $$C:3$$ -Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $$C:{\ell }$$ -Contractibility is $$\textsf{NP}$$ -complete on bipartite graphs for $$\ell = 6$$ and posed as open problems the status of $$C:{\ell }$$ -Contractibility when $$\ell $$ is 4 or 5. In this paper, we show that both $$C:5$$ -Contractibility and $$C:4$$ -Contractibility are $$\textsf{NP}$$ -complete on bipartite graphs. © 2022, Springer Nature Switzerland AG.