Summary: Let \(\mu\) be a positive measure on \([-1,1]\) satisfying the Szegő condition \[ \int^1_{-1} \frac {\log \mu'(t)}{\sqrt{1-t^2}}dt >-\infty \] with \(\mu'\) being the Radon-Nikodým derivative with respect to the Lebesgue measure, and let further \(w_{2n}\in \mathcal P_{2n}\), \(n\in \mathbb N\), be a sequence of polynomials of degree at most \(2n\), with zeros \(a_{2n,1},\dots,a_{2n,m}\in \mathbb C\setminus[-1,1]\), \(m=\operatorname{deg}(w_{2n})\), and being positive on \([-1,1]\). We study the asymptotic behavior of the polynomials \(p_n=p_n(\mu_n;\cdot)\) orthonormal with respect to the varying measure \(d\mu_n:=w^{-1}_{2n}d\mu\). Let \(\varphi:\overline{\mathbb C}\setminus [-1,1]\to\mathbb D\) be the conformal mapping of \(\overline{\mathbb C}\setminus[-1,1]\) onto \(\mathbb D\) with \(\varphi(\infty)=0\) and \(\varphi'(\infty)>0\). Under the assumption that \[ \lim_{n\to\infty}\Biggl[\big(2n-\operatorname{deg}(w_{2n})\big)\sum^{\operatorname{deg}(w_{2n})}_{j=1} (1-|\varphi(a_{2n,j})|)\Biggr]=\infty \tag \(*\) \] we prove strong asymptotic relations in \(\mathbb C\setminus [-1,1]\) for the orthonormal polynomials \(p_n(\mu_n;\cdot)\) as \(n\to \infty\). An analogue of this result is proved for polynomials that are orthonormal with respect to varying measures on the unit circle \(\mathbb T=\partial \mathbb D\). Also in this case the original measure of orthogonality has to belong to the Szegő class (on \(\mathbb T\)) and an assumption analogous to \((*)\) has to hold true. The necessity of this assumption will be discussed in some detail for the case of orthogonality on \(\mathbb T\). The strong asymptotic relations for \(p_n(\mu_n;\cdot)\) lead to correspondingly precise asymptotic error estimates for multipoint Padé approximants (i.e., rational interpolants) to Markov functions \(f(z)=\int(t-z)^{-1}d\mu(t)\). The error estimates for multipoint Padé approximants have been the primary motivation for studying strong asymptotics in the present paper.