A Turkish official said much of the material in question was related to the Kurdistan Workers Party (PKK) which Ankara deems as a terrorist organisation.
A spokesperson for Twitter confirmed the company has taken legal action over the fine without providing further details.
Ankara has taken a tough stance on social media under President Tayyip Erdogan and the ruling AK Party he founded. It has temporarily banned access to Twitter site several times in the past for failing to comply with requests to remove content.
But the 150,000 lira ($50,000) fine, imposed by the BTK communications technologies authority, was the first of its kind by Turkish authorities on Twitter.
Twitter, in its lawsuit, is arguing that the fine is against the law and should be annulled, the source said.
The content Turkish authorities have asked to be removed includes tweets in relation to the PKK, which is also considered a terrorist organisation by the European Union and the United States, a Turkish official said. Some tweets are related to the far-left DHKP-C.
"We have shown 15-20 tweets from several accounts to Twitter as examples. We have imposed the fine because Twitter failed to comply with the court order," this official said.
Transport Minister Binali Yildirim said on Wednesday that Turkey would not give up on its demand for Twitter to pay the fine.
The government has also introduced legislation making it easier for such bans to be imposed. Turkey is among the top countries with the highest number of content-removal requests to Twitter, data from U.S.-based company shows.