How to eradicate excess lines in a singly recorded polygonal line (road) network? QGIS 2.8

I’m a beginner in GIS software.

I’m trying to create a road network shapefile within the reserve I work at. To do this, I’ve driven each and every road inside the property with my GPS app set to record the track. I’ve exported this track into QGIS, saved it as an ESRI shapefile, and extracted the nodes. My ultimate aim is to use this line network to create polygons within the boundaries of surrounding and intersecting lines to subdivide the reserve into sections for easier management. You can see what I’ve done so far below:

My problem is the following — although I did my best during recording, there were a number of single roads that I recorded more than once (i.e I drove some roads, with the GPS app tracking, 2 or more times each when for example I had to follow a loop road). The result is that instead of having line intersections only at the actual junctions as per satellite/ real-time view, I’m left with this chaotic mish-mash of intersections as you can see below:

This makes it impossible for me to draw out clearly defined polygons from these lines, especially as some lines are just short of intersecting as a result of me pausing and unpausing the track. To add to my woes, the track I recorded has no attribute data (i.e order of nodes) that I can use to break the whole network into segments and connect individually using the ‘Join Multiple Lines’ function. Because of this, trying to use Geoprocessing functionalities such as ‘Dissolve’ or ‘Eliminate Sliver Polygons’ is futile

How do I merge or remove these excess lines and make one completely linear, 2D shapefile of the road network I can use to make clear polygons, considering the above? Please note that I am still using QGIS 2.8.2 and for various reasons am not considering upgrading

I hope I’ve explained the problem adequately and I have a deadline to follow for this task.