The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power 𝛼. Using a dynamical version of the strong disorder renormalization group we find for 𝛼>𝛼𝑐 that the entanglement entropy grows logarithmically with time and becomes smaller with larger 𝛼 as 𝑆⁡(𝑡)=𝑆𝑝⁡ln⁡(𝑡)/(2⁢𝛼). Here, 𝑆𝑝=2⁢ln⁡2−1. We present results of numerical exact diagonalization calculations for system sizes up to 𝑁∼16 spins, in good agreement with the analytical results for sufficiently large 𝛼>𝛼𝑐≈1.8. For 𝛼<𝛼𝑐, we find that the entanglement entropy grows as a power law with time, 𝑆⁡(𝑡)∼𝑡𝛾⁡(𝛼) with 0<𝛾⁡(𝛼)<1 a decaying function of the interaction exponent 𝛼.