Did teenage Sultan Osman II use people as targets?

SHORT ANSWER

It's impossible to be certain but none of the sources which mention this story cite a source. Further, there are inaccuracies in some of the other information they give on Osman II. Also, none of the more detailed accounts of Osman II by the modern historians listed at the end of this post mention this sultan using prisoners of war and pages for target practice.

DETAILED ANSWER

The story you cite appears in Volume X of The Story of the Greatest Nations (1914). On Osman II, the authors say:

He was a savage youth who practised archery by shooting at prisoners of war, and when the supply of these ran low, he fastened up one of his own attendants as a target.

This story is also mentioned on Mad Monarchs (2011) in the bio of Mustafa I, Osman's uncle:

Young Osman was very fond of archery - especially with living targets, like prisoners-of-war or his own pages.

and again in The Rough Guide to Istanbul (2012)

Osman II...enjoyed archery, but only when using live targets, including prisoners of war and his own pages.

C.S. Denton, in Absolute Power: The Real Lives of Europe’s Most Infamous Rulers (2006) and in Ruthless Rulers: The Real Lives of Europe's Most Infamous Tyrants (2016) also mentions Osman II is his section on Mustafa I:

Only two things occupied Osman's mind: his archery skills, which he liked to practice with his own pages and prisoners of war as targets, and his religion, which him feel compelled to try to reform the Janissaries,...

At first glance, five sources seems to be evidence of some substance. However, none of them cite a specific source for this story. Further, two of the above are by the same author (Denton), one (Greatest Nations) dates from 1914 and has neither footnotes nor bibliography; one of the authors wrote primarily fiction while the other was a Professor of English, not history. Another source, Rough Guide, is not a historical source. Mad Monarchs seems the most credible and gives a list of sources at the end (but no specific citations), even though it was not written by a professional historian.

There are other problems with four of the above sources (make that three as the texts in Denton's two books are identical). For example, Rough Guide implies that Osman II was "crazed" while Denton says that he was almost as deranged as his uncle Mustafa I. None of the sources by professional historians cited below (see list) back up this assertion of madness, and many of them go into far more detail on the life and reign Osman II than any of sources claiming he used prisoners and pages for target practice. In fact, the picture that emerges of Osman II is one of a young sultan with a clear agenda and definite ideas on how to revive Ottoman fortunes, but whose plans threatened too many vested interests, leading to his overthrow and death by strangulation.

Also, Greatest Historians on Osman II is highly critical in every respect, perhaps reflecting the view of most Ottoman historians of the time who did not approve of the young sultan's radical plans to revive Ottoman fortunes.

None of this is to say that Osman II wasn't ruthless on occasion; he did have one of his brothers murdered (fratricide was a practice which was not without precedence among the Ottomans - see this and this), and who is to say that he didn't loose an arrow in anger at some point, but that is a long way from saying he used people for target practice.

Wikipedia mentions a similar story to the one in the question title about one of Osman's successors, Murad IV (sultan 1623 to 1640):

...he would sit in a kiosk by the water near his Seraglio Palace and shoot arrows at any passerby or boatman who rowed too close to his imperial compound, seemingly for sport.

Other sources consulted:

The Climate of Rebellion in the Early Modern Ottoman Empire (2011)

Rumor and Regicide (2010)

An Ottoman Tragedy: History and Historiography at Play (2003)

Osman's Dream: The History of the Ottoman Empire (2007)

Erdal K.Yalcin, Who Killed Sultan Osman? (2003 or later)

Mehrdad Kia, The Ottoman Empire (2008)

II. Osman