The most promising mechanisms for the formation of Majorana bound states (MBSs) in condensed matter systems involve one-dimensional systems [such as semiconductor nanowires, magnetic chains, and quantum spin Hall insulator (QSHI) edges] proximitized to superconducting materials. The choice between each of these options involves tradeoffs between several factors such as reproducibility of results, system tunability, and robustness of the resulting MBS. In this paper, we propose that a combination of two of these systems, namely, a magnetic chain deposited on a QSHI edge in contact with a superconducting surface, offers a better choice of tunability and MBS robustness compared to magnetic chain deposited on bulk. We study how the QSHI edge interacts with the magnetic chain, and see how the topological phase is affected by edge proximity. We show that MBSs near the edge can be realized with lower chemical potential and Zeeman field than the ones inside the bulk, independently of the chain's magnetic order (ferromagnetic or spiral order). Different magnetic orderings in the chain modify the overall phase diagram, even suppressing the boundless topological phase found in the bulk for chains located at the QSHI edge. Moreover, we quantify the quality of MBSs by calculating the Majorana polarization (MP) for different configurations. For chains located at the edge, the MP is close to its maximum value already for short chains. For chains located away from the edge, longer chains are needed to attain the same quality as chains located at the edge. The MP also oscillates in phase with the in-gap states, which is relatively unexpected as peaks in the energy spectrum correspond to stronger overlap of MBSs.