During bridge amplification, when sequences attached to adapters on the surface form "bridges" and are replicated, it seems like sequences with either end attached to the surface will be created. For instance, in this link in figure 7, there are sequences facing both ways on the plate. In step 8, all the sequences are oriented in the same direction (pink end attached to the surface) How are does this occur? How are the other sequences removed?
See this document on how Illumina sequencing works.
When you are doing a single end sequencing, the reverse strands produced after the bridge amplification + denaturation (step-7) are specifically removed (clipped at the junction of primer and the "insert" region). Furthermore, the free 3' ends are blocked (by chemical modification) to ensure that there is no unwanted priming.
For paired end sequencing, both strands are sequenced; so in one flow cell the forward strand is cleaved while in the other the reverse strand is cleaved.
Presumably they removed one set of fragments from their figures for clarity of demonstration. In reality, both fragments remain immobilized on the chip. The sequencing primer only anneals to one adapter which allows selective extension of one of the sets in each run. For fragments of sufficient length, this provides an easy mechanism for creating paired end sequence reads as you can use one primer to read one strand and, subsequently, another primer to read the other strand.