A way to search street intersection on openstreetmap

You could use the following approach:

[bbox:{{bbox}}];
way[highway][name="6th Avenue"];node(w)->.n1;
way[highway][name="West 23rd Street"];node(w)->.n2;
node.n1.n2;
out meta;

Try it on overpass turbo: http://overpass-turbo.eu/s/6Pb

Edit: node.n1.n2; calculates the intersection of input set .n1 and .n2. Please check the documentation for details.

Works fine for me. Could be enhance.

[bbox:{{bbox}}];
(way[highway="motorway"];way[highway="trunk"];way[highway="primary"];way[highway="secondary"];way[highway="tertiary"];way[highway="unclassified"];way[highway="track"];way[highway="service"];way[highway="residential"];)->.n1;
foreach.n1(
    (.n1; - ._;)->.n2;
    node(w._)->.n3;
    node(w.n2)->.n2;
    node.n3.n2;
    out;
  );

http://overpass-turbo.eu/s/bkD

It keeps crossing, if you don't need it, just substract the set of crossing to the intersection node (node.n3.n2).