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In the present paper we extend the L² Korn interpolation and second inequalities in thin domains, proven in [D. Harutyunyan, SIAM J. Math. Anal., 50 (2018), pp. 4964-4982], to the space Lp for any 1 < p < ∞ . A thin domain in space is roughly speaking a shell with nonconstant thickness around a smooth enough two dimensional surface. The inequality that we prove in Lp holds for practically any thin domain Ω \subset R³ and any vector field \bfitu \in W1,p(Ω). The constants in the estimate are asymptotically optimal in terms of the domain thickness h. This in particular solves the problem of finding the asymptotics of the optimal constant in the classical Korn second inequality in Lp for thin domains in terms of the domain thickness in almost full generality. The remarkable fact is that the interpolation inequality reduces the problem of estimating the gradient \nabla μ in terms of the strain e(μ) to the easier problem of estimating only the vector field μ in terms of e(μ), which is a Korn--Poincaré inequality. [ABSTRACT FROM AUTHOR]