We have fabricated and tested a simple circuit of the rapid single-flux-quantum(RSFQ) four-stage shift register using a single layer high-$T_c$ superconducting (HTS) $YBa_2Cu_3O_{7-x}$ (YBCO) thin film structure with 9 step-edge Josephson junctions. The circuit includes two read superconducting quantum interference devices(SQUID) and four stages. To establish a robust HTS RSFQ device fabrication process, we have focussed on the reproducible process of sharp and straight step-edge formation as well as the ratio of film thickness to step height, t/h. The spread of step-edge junction parameters was measured from each 13 junctions with t/h=1/3, 1/2, and 2/3 at various temperatures. We have demonstrated the simplified operation of the shift register at 65 K.