Issue with linking lines?

What Jacob said.

For that to be a logo you would later need to outline the strokes. Then you can unite all the shapes.

Could you please show the screenshot of the Outline mode (Ctrl/Cmd + Y)?
There you can see the real shapes, not disturbing by their appearance.
If they are lines, it may be needed to check if they are in different groups and ungroup them. If, au contrary, they are other shapes (e.g. rectangles), then youwill have to move their edges to overlap each other and use Pathfinder/Unite. But in this last case you will need to convert all lines to shapes (or redraw these bottom lines as lines, not shapes).

hernameisjack,

I believe you have two issues.

You can only join open end Anchor Points.

To have a right appearance of the outer bounds of the shape, you need to create it as one closed path and use open paths inside it, unless you use Round Joins.

I would suggest a remake where you create the closed outer path, possibly starting by joining the separate open 2 point paths at the bottom, then trying to join further round, and opening the closed paths you come across, which can be done with the Scissors Tool or by deleting the segments following the outer bounds.

Each of the longer straight parts of the outer bounds (in the middle and at the top) can be made as a single path/segment, just passing the ends of the inner parts.