In the beginning Batman's appearances in comic books are limited to short stories, between six and fifteen pages long. Starting in Detective Comics in May of 1939, each issue featured one Batman story placed first before a range of other material. The Batman comic book started in the spring of 1940, with each issue featuring four short Batman and Robin stories. Batman's first appearance in Detective Comics #27 is six pages long, and the four stories that made up a Batman comic of the era would each be about thirteen pages. The shorter, early Detective Comics would feature about eight or nine panels per page, while Batman comics would feature six or seven.
Most of the original Batman stories are self standing, the longest running narratives run across two stories, usually in consecutive issues.
The first bit of this material will cover the first two distinct phases of Batman comics. My understanding is that all pre-1946 Batman would be considered by definition 'Golden Age'. I am not a comic historian in any sense, but anyone can see a distinct style change that took place in the Golden Age comics that is worth noting for visualization and analysis.
May of 1939 to March of 1940 is the original Batman who worked independently, was more violent, and had a tenuous relationship with police, but was on good terms with Commissioner Gordon. (Commissioner Gordon is in the very first appearance of Batman, and he pre-dates all other Batman friends or foes) This Batman wears black. Updated versions of this Batman are in Tim Burton's Batman film, The 2004 The Batman Saturday morning cartoon and the recently released First Wave comic books. One of the most interesting things about the 'original' Batman is that he only was in this state for 11 monthly issues of Detective Comics, with the first stories being six pages long, and eventually growing to 12. There are only 110 pages of this material.(with a possible addition of 12 more, see below) These stories are often of business deals gone bad, mad scientists, and werewolves. They could potentially be considered thriller/horror in terms of genre.
April of 1940 is when Robin, the Boy Wonder appears, and sets the tone for World War II. His appearance directly coincides with World War 2 escalating in Europe. Batman becomes less violent, and has a very positive relationship with police. This Batman wears blue. As soon as Robin appears the tone of the comics changes drastically. The comics become increasingly light in tone, and the colorful Batman Rogues appear. The costumed hero versus costumed villain pastiche begins to appear clearly as these comics progress.Within six months of his introduction there are more pages of Batman & Robin than of Batman's previous solo comic.
Although it is not a perfect split, the change in the comics happens with the appearance of Robin, in Detective Comics #38 and Batman #1.The second story from Batman #1does not include Robin, and this is the story advertised as the next issue of Detective Comics in Detective Comics #37 (it's also the only episode where Batman kills people). By appearances the third and final appearance of Dr. Hugo Strange in the Golden Age should be considered the last original black Batman. It is also not a complete tonal change. Some of the early comics including Robin could be fairly dark, in Batman #6 Robin wrestles a guy into shooting himself in the head.
Thus for expediency April 1940 will be used to designate 'The Batman' (pre April 1940) from 'The Dynamic Duo' (post April 1940).
Long story short, if you want to know when the original scary Batman we all have heard about was phased out for happier Batman, it was when Robin arrived in April 1940.