Per the comments, and as Erica originally suggested, your network isn't correctly set up for analysis as evidenced by a severe lack of junctions. This could result from either geometry issues with the data or how the connectivity policy is defined. I suspect both might be an issue in your case, as most junctions I see are at line endpoints, but there are some at apparent intersections and some endpoints do not have junctions. Your service area polygons aren't spread out because the analysis cannot reach those areas of the network.
The geometry issue: In a network when two lines cross that doesn't mean there is actually an intersection or junction. A turn (or transfer in multi-modal networks) can only be made at a junction. In order for a junction to occur both lines must have a vertex at a common point. Even if one line has a vertex snapped to the other edge, it won't work - they both have to have one. These issues must be addressed in the source line data from which the network is created.
The connectivity policy issue: The connectivity policy is a network dataset property and can be set one of two ways related to this issue.
- The connectivity policy for the street layer is set to End only: a junction can only occur at a line endpoint (aka node, or first/last vertex).
- The connectivity policy for the street layer is set to Any Vertex:
Junctions can occur at any vertex, be they at the beginning or end of the line or anywhere in between.
This network connectivity policy is easiest demonstrated by creating a T intersection. Create a line with three vertices, then a perpendicular line with two - one of which snaps to the middle vertex of the first line. Now create your start point on one leg of the T and your stop on another. In case 1 above, the route will fail with an error message that the destination is unreachable because no connection can be made at the middle of the line. In case 2 the route will succeed, because a junction is allowed at a non-nodal vertex.
A functional network of the area you're looking at should appear more like this: