How did American colonies achieve democratic government?

The American colonies never achieved democratic government. They made great strides towards it and laid the foundation for the democratic republic formed by the United States Constitution, but the colonies were quite far from proper democracies when they finally declared independence in the Revolutionary War.
The path towards democratic government in the colonies began in Plymouth, Massachusetts in 1620. The Pilgrims, bound for the Virginia colony, found themselves instead in Massachusetts. Without colonial government or royal charter to govern them, they had to form their own civic body to manage affairs in their small community. They agreed to do so in the Mayflower Compact, the first American declaration of self-government.
Royal rule would eventually spread to Massachusetts, which was granted a Charter and a great deal of freedom to continue to self-govern. Rhode Island and Connecticut were granted similar latitude. However, the remaining ten colonies were incorporated as Royal or Proprietary colonies, led by individual governors who reported directly to the King. All thirteen colonies had had elected legislatures, but the Governor and the King both had absolute veto rights over any laws passed by most of these legislative bodies.
The Mayflower Compact was short-lived, but the spirit of freedom that it fostered would last much longer. Colonial legislatures chafed under the undemocratic restrictions. The developed workarounds, such as passing and re-passing short-term legislations that would expire and be renewed before they could even reach the King for veto--but this still left the colonies beholden to the King's appointed governor. Moreover, voting for colonial legislators was extremely limited. In the early days of the nation, the United States profoundly limited itself to a white male electorate; but in the days of the Colonies, voting was a privilege held only by white male taxpaying, churchgoing landowners.
When the War of Independence began, the states convened their elective representatives to form a Continental Congress and agree to the terms of their Union. The result of this negotiation were the Articles of Confederation, a loose framework that defined the several States' relationships with one another. This began the notion of a Federal government subsuming some, but not all, of the responsibilities that would normally fall to the Colonies or States themselves. Yet this rudimentary document made no mention of elections or voting; it was not until Independence was won and the Constitution drafted that democratic principles were formally enshrined in American law.
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