Learn Turkish: Sinsi kadını sokakta beklerken gördüm. - I saw the sneaky woman while she was waiting in the street.

Good catch! The suffix -(y)ken doesn't take personal endings. Usually, an action with -(y)ken is performed by the subject of the main verb (which is "I" here). However, with verbs of perception or discovery (like görmek, bulmak, or yakalamak), -(y)ken often modifies the direct object. Because sinsi kadını (the sneaky woman) is the accusative object leading into the action, it naturally reads as "I saw her [in a state of] waiting".

How would I say "I saw the sneaky woman while I was waiting in the street" instead?

To make it clear that you were the one waiting, you would change the word order to attach the -(y)ken phrase to the subject, or move the object closer to the main verb: Sokakta beklerken sinsi kadını gördüm. or Ben sokakta beklerken sinsi kadını gördüm. Putting the object (kadını) after the waiting action disconnects her from it.

Why is it beklerken and not bekliyorken?

You will actually hear both! While bekliyorken (using the present continuous -iyor) strongly emphasizes an ongoing action right at that exact second, using the aorist (-r / -ar / -er) with -(y)ken is the standard, most idiomatic way to say "while doing something." Both translate perfectly to "while waiting," but beklerken flows a bit more naturally in everyday speech.