During bridge amplification of DNA sequences, why aren't sequences amplified in both orientations?
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.