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Let $G$ be a real semisimple Lie group with trivial centre and no compactfactors. Given a conjugate pair of either real hyperbolic elements or unipotentelements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that$d_G(1,g) \leq L(d_G(1,u)+d_G(1,v))$, where $L$ is a positive constant whichwill depend on some property of $a$ and $b$. For the vast majority of suchelements however, $L$ can be assumed to be a uniform constant.

Bounded Conjugators For Real Hyperbolic and Unipotent Elements in Semisimple Lie Groups