ISTANBUL—Turkey's fractured opposition has been unable to capitalize much on the biggest outpouring of public anger in decades, while Prime Minister Recep Tayyip Erdogan has exploited their divisions to strengthen his personal standing over the three weeks of street protests.

The mostly educated, middle-class protesters have been united by a rising distaste for the governing Justice and Development Party, or AKP, and what they say is Mr. Erdogan's increasingly heavy-handed rule and encroachment on civil liberties since easily...