If I try the “proper” way, that is, making the apostrophe a letter as far as hyphenation is concerned, I get
İs-tan-bu-l-’-lu-la-rın-ki
because the patterns allow hyphens at either side of a consonant except for some specific combinations, which don't take the apostrophe into consideration. Perhaps you would consider making a feature request at the TeX-hyphen mailing list.
The “proper” way would be to set, for Turkish, \lccode"2019="2019 (which is the magic making the apostrophe to be considered for hyphenation).
One has to resort to babel style shorthands:
\documentclass{article}
\usepackage{polyglossia}
\usepackage{testhyphens}
\setmainlanguage{turkish}
\makeatletter
\ifcsundef{initiate@active@char}{%
\input{babelsh.def}%
\initiate@active@char{'}%
}{}
\def\turkish@shorthands{%
\bbl@activate{'}%
\def\language@group{turkish}%
\declare@shorthand{turkish}{'}{\turkish@apostrophe}%
}
\def\noturkish@shorthands{%
\@ifundefined{initiate@active@char}{}{\bbl@deactivate{'}}%
}
\define@boolkey{turkish}[turkish@]{babelshorthands}[true]{}
\setkeys{turkish}{babelshorthands=true}
\appto\blockextras@turkish{\ifturkish@babelshorthands\turkish@shorthands\fi}
\appto\inlineextras@turkish{\ifturkish@babelshorthands\turkish@shorthands\fi}
\appto\noextras@turkish{\noturkish@shorthands}
\newcommand\turkish@apostrophe{\@ifnextchar'{”\@gobble}{’\hspace{0pt}}}
\makeatother
\begin{document}
\begin{checkhyphens}{}
İstanbullularınki
İstanbul'lularınki
\end{checkhyphens}
Double quotes are OK: \fbox{''}
\end{document}

This allows a line break after the apostrophe; if you don't want it, then use
\nobreak\hspace{0pt}
Note that this doesn't affect the “real” apostrophe (it can't be used as a shorthand). If you are using ’ in your document, then you can exploit \newunicodechar:
\documentclass{article}
\usepackage{polyglossia}
\usepackage{testhyphens}
\setmainlanguage{turkish}
\makeatletter
\ifcsundef{initiate@active@char}{%
\input{babelsh.def}%
\initiate@active@char{'}%
}{}
\def\turkish@shorthands{%
\bbl@activate{'}%
\def\language@group{turkish}%
\declare@shorthand{turkish}{'}{\turkish@apostrophe}%
}
\def\noturkish@shorthands{%
\@ifundefined{initiate@active@char}{}{\bbl@deactivate{'}}%
}
\define@boolkey{turkish}[turkish@]{babelshorthands}[true]{}
\setkeys{turkish}{babelshorthands=true}
\appto\blockextras@turkish{\ifturkish@babelshorthands\turkish@shorthands\fi}
\appto\inlineextras@turkish{\ifturkish@babelshorthands\turkish@shorthands\fi}
\appto\noextras@turkish{\noturkish@shorthands}
\newcommand\turkish@apostrophe{\@ifnextchar'{”\@gobble}{\string’\hspace{0pt}}}
\makeatother
\usepackage{newunicodechar}
\newunicodechar{’}{'}
\begin{document}
\begin{checkhyphens}{}
İstanbullularınki
İstanbul'lularınki
İstanbul’lularınki
\end{checkhyphens}
Double quotes are OK: \fbox{''}
\end{document}

polyglossiapackage do you have? (The version that's distributed with MacTeX2014 is v1.33.5, dated 2014/05/21. When I run your MWE on my system, I do not encounter the problem you report getting.) – Mico Mar 23 '15 at 13:38polyglossiapackage appears to be the same as yours: 2014/05/21 v1.33.5. – Caleb Mar 23 '15 at 13:47İs-tan-bu-l-’-lu-la-rın-ki, because the patterns allow hyphens at either side of a consonant except for some specific combinations, which don't take the apostrophe into consideration. – egreg Mar 23 '15 at 16:50