No problem... Try running the Integrate geoprocessing tool--assuming you have the correct license level. This will create vertices where lines are touching.
The issue is that your lines may be touching one another, but in order to connect in the network, the lines need to touch at vertices or endpoints. The attached graphics show what I mean.
1_Without_Vertices.png is what you have currently. The blue dot (hard to see under the selected line) represent an endpoint, the green squares show vertices. There is an endpoint on one line, but there isn't an endpoint or a vertex on the line it touches. The resultant network won't connect there.
2_With_Vertices.png is what the street features look like after running Integrate. When I create a network with Any Vertex connectivity, the route can turn there.
By the way, you don't have to run Integrate. It's just a quick way to add vertices where features touch one another. You can add vertices and endpoints through editing or other geoprocessing tools.
Robert