Well, first off you are going to have difficulty with the term conservative, since it isn't well defined.
Conservatism (Latin: conservare, "to retain") is a political and social philosophy that promotes retaining traditional social institutions. A person who follows the philosophies of conservatism is referred to as a traditionalist or conservative. [...] The term, historically associated with right-wing politics, has since been used to describe a wide range of views. There is no single set of policies that are universally regarded as conservative, because the meaning of conservatism depends on what is considered traditional in a given place and time.
So, maybe you just meant, "Whatever the Republican Party believed at the time." If so, there have been several periods of time of sustained "conservatism."
For starters, the Republican Party was formed from the defunct Whig party (possibly the original conservatives), to oppose the Southern Democrats who wished to continue the social institution of slavery. I guess this begs the question, who were the conservatives?
Founded by anti-slavery activists in 1854, it dominated politics nationally for most of the period from 1860 to 1932. There have been 18 Republican presidents, the first being Abraham Lincoln, serving from 1861-1865
Founded in the Northern states in 1854 by anti-slavery activists, modernizers, ex-Whigs and ex-Free Soilers, the Republican Party quickly became the principal opposition to the dominant Southern Democratic Party [...] The Northern Republicans saw the expansion of slavery as a great evil. The first public meeting where the name "Republican" was suggested for a new anti-slavery party
The Republican Party dominance lasted until the repeal of Prohibition. Both Democrats and Republicans helped pass Prohibition, which was a major cause of the Progressive Movement, but the wet Democrats repealed it. The other major Progressive cause was Woman's Suffrage, that the Republicans heavily supported.
On May 21, 1919, it was passed, 304 to 89, (Republicans 200-19 for, Democrats 102-69 for, Union Labor 1-0 for, Prohibitionist 1-0 for), 42 votes more than necessary being obtained. On June 4, 1919, it was brought the Senate, and after a long discussion it was passed, with 56 ayes and 25 nays (Republicans 36-8 for, Democrats 20-17 for)
Once again, you have a question of who the conservatives are? The Progressives that wanted the radical change in voting, and who wanted moral and economic control of all Americans to bring about the great social revolution?
Progressivism was, to a great extent, the culmination of the pietist Protestant political impulse, the urge to regulate every aspect of American life, economic and moral – even the most intimate and crucial aspects of family life. But it was also a curious alliance of a technocratic drive for government regulation, the supposed expression of "value-free science," and the pietist religious impulse to save America – and the world – by state coercion.
Pietists and "progressives" united to control the material and sexual choices of the rest of the American people, their drinking habits, and their recreational preferences. Their values, the very nurture and education of their children, were to be determined by their betters. The spiritual, biological, political, intellectual, and moral elite would govern, through state power, the character and quality of American family life
Then you have the Civil Rights Act of 1964, which lead the Republicans into dominating the south after the Jim Crow Democrats were unable to block Republican efforts to bring equality to blacks in the south.
The Original House version:
Democratic Party: 152–96 (61–39%)
Republican Party: 138–34 (80–20%)
Cloture in the Senate:
Democratic Party: 44–23 (66–34%)
Republican Party: 27–6 (82–18%)
The Senate version:
Democratic Party: 46–21 (69–31%)
Republican Party: 27–6 (82–18%)
The Senate version, voted on by the House:
Democratic Party: 153–91 (63–37%)
Republican Party: 136–35 (80–20%)
So, who are the conservatives?
Without knowing what you mean by conservatives, it is difficult to determine how long conservatism was sustained.