I read the times almost everyday and have plenty of conservative opinions. The Times has multiple conservative columnists and consistently publishes conservative op-eds.
These 'conservative' columnists do not reflect the majority of conservative opinion in this country though. NYT columnists tend to be members of the political or cultural elite. There is a large strain of anti-elite sentiment running through the country right now, especially on the right. These people do not feel their views are expressed in the NYT. For that matter, I know plenty of working-class dems who feel disenfranchised with the NYT, the media elite, the democratic party, etc.
Conservatives at NYT do not reflect the opinions of most conservatives or most conservative media.
If you want a taste of conservative media, look at what is collated on RealClearPolitics. NYT is very different, you won't see Ross Douthat saying the same things they say in The Federalist.