Chromosomes in the eukaryotic nucleus are highly compacted. However, for many functional processes, including transcription initiation, the pairwise motion of distal chromosomal elements such as enhancers and promoters is essential and necessitates dynamic fluidity. Here, we used a live-imaging assay to simultaneously measure the positions of pairs of enhancers and promoters and their transcriptional output while systematically varying the genomic separation between these two DNA loci. Our analysis reveals the coexistence of a compact globular organization and fast subdiffusive dynamics. These combined features cause an anomalous scaling of polymer relaxation times with genomic separation leading to long-ranged correlations. Thus, encounter times of DNA loci are much less dependent on genomic distance than predicted by existing polymer models, with potential consequences for eukaryotic gene expression.