Was there a battle in history where militia was effective?

A clear case would be the Battle of New Orleans during the war of 1812. The American forces included about a dozen regiments of various southern state militias. There is a very good order of battle referenced on wikipedia

Columbus, New Mexico. Pancho Villa was crossing over the border and raiding on the American side. He hit Columbus and ran into a National Guard unit and bunch of locals with guns. They shot the hell out of his bandits. Northfield, MN. The James Gang went into the bank to rob it. Locals saw him, armed themselves and when his gang came out they got the hell shot out of them. If your are looking for a bigger action, the first troops of the USA were nothing but militia, at Lexington and Concord. In the battle of Concord the British had left Boston to seize an arms cache in Concord. They were met by a quickly armed militia in Lexington. They continued onto Concord, holding a key bridge. They were driven back when the militia did a volley fire on them. The British began a retreat and more and more militia came after them. The British started abandoning equipment so they could retreat faster. (Run faster.) The British continued to retreat until they got to a place where they had naval support. I would call that a win for the militia and they had minimal if any training before the fight. Washington in fact had to hire a German officer to train his army. As far as today, the Geneva Conventions recognizes THREE components of a nations military forces. The Regular (full time) forces, the Reserve (Reserve and Guard) and levee en masse. (Civilians taking up arms to drive out an invader.) In other words, the last is the classic idea of a civilian militia. These conventions were signed after WW TWO, in 1949.

Depends what you mean by "militia". Note that in many parts of the world for much of history there were no organised "armies" as we know them in modern times in countries such as the USA and the UK and France. Thus many battles were fought (at least mostly) by "part-time soldiers" on both sides. Some of those "part-time soldiers" might well answer to the description of "militias" that you are asking about.

Note that I am writing about time well before the establishment of the USA, and indeed before European knowledge of the Americas.

Note that your "watching" is watching modern reconstructions of battles, which might have biased coverage. There was no moving image recording before about 1890 - thus WW1 was the first big war in which real movie footage was recorded on the battlefield. Even still photography is less than 200 years old, and could record only static scenes in its first few decades, thus still photos from the American Civil War are "posed".

Paintings of battle scenes in earlier wars were usually from the victor's point of view, and were painted weeks or months after the battles themselves. And remember the old saying, that history is written by the victors.

At the Battle of Stamford Bridge.

Kokoda track where 1500 Australian Militia stopped the Japanese from getting over the Owen Stanley Ranges to invade Australia all without any Help from the USA

One Japanese reported unable to cross over range Position being Protected by 2,000 Australian Troops well dug in the Truth it was only 162 Australians dug in and all Militia

The Battle of Marathon

Bunker Hill 1775 was good. The USA Colonial militia held the hill and fought the British regulars/ professionals/ mercenaries to a standstill. The British did eventually takes the hill but the British suffered 3 times as many casualties as did the militia, and therefore crippled themselves for further movement.

Militias have been used throughout history for thousands of years. Typical militia were just civilians of a besiged town or city that were drafted into service to help defend. Other militias were used in attacking as well to help bolsters an armies numbers. Militias were given some form of training, even in the past. At the same time militias could of been given no training at all. Typically when helping defend a city militias were given some form of spear like weapon, and the most bare minimum training on how to use it and what tactics to do. Usually how the should stand togeather, how to hold the spear, how to attack with it, and what they need to do. This type of training can be a about an hour or possibly a few days or weeks long. Depending how long the city was under siege.

Every war up until WWI when the US created a standing army.

Militia units were used during the war of 1812, many of the units who fought in the US Civil war were Militia units. The US continued to depend on militia units through at least the Spanish American war.

As support information consider the names of the units that fought these wars/battles, more then one unit had a name associated with a state.