How is it possible that regions thousands of base pairs from a promoter can impact transcription of that gene?

DNA can make loops

Explanation:

Transcriptional factors (such as activators or repressors) are proteins that regulate gene expression by binding to DNA sequence (such as enhancers and silencers). Consequently, gene transcription might be turned on or off. Usually binding sites for transcriptional factors are located near the promoter (initiation of transcription). But when they are located far from the gene they regulate DNA, flexibility plays a role. DNA can form loops which bring together binding sites and transcription factors.