We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number $c$ and the weak closure number $γ$ [Fox et al., SICOMP 2020] in addition to the standard parameter solution size $k$. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size $k^{\mathcal{O}(γ)}$ and $(γk)^{\mathcal{O}(γ)}$, respectively, thus extending previous kernelization results on degenerate graphs. The kernels are essentially tight, since these problems are unlikely to admit kernels of size $k^{o(γ)}$ by previous results on their kernelization complexity in degenerate graphs [Cygan et al., ACM TALG 2017]. In addition, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number~$c$ and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.