Context. Until now, there have only been seven quasars at z > 4.5 whose the high-resolution radio structure had been studied in detail with Very Long Baseline Interferometry (VLBI) imaging. Aims. We almost double the number of VLBI-imaged quasars at these high redshifts with the aim of studying their redshift-dependent structural and physical properties in a larger sample. Methods. We observed five radio quasars (J0813+3508, J1146+4037, J1242+5422, J1611+0844, and J1659+2101) at 4.5 < z < 5 with the European VLBI Network (EVN) at 1.6 GHz on 29 October 2008 and at 5 GHz on 22 October 2008. The angular resolution achieved ranges from 1.5 to 25 milli-arcsec (mas), depending on the observing frequency, the position angle in the sky, and the source's celestial position. Results. The sources are all somewhat extended on mas scales, but compact enough to be detected at both frequencies. With one exception of a flat-spectrum source (J1611+0844), their compact emission is characterised by a steep radio spectrum. We found no evidence of Doppler-boosted radio emission in the quasars in our sample. The radio structure of one of them (J0813+3508) is extended to ∼7 , which corresponds to 43 kpc projected linear size. Many of the highest redshift compact radio sources are likely to be young, evolving objects, far-away cousins of the powerful gigahertz peaked-spectrum (GPS) and compact steep-spectrum (CSS) sources that populate the Universe at lower redshifts.