A turbine shroud assembly includes a plurality of arcuate shroud block assemblies annularly arranged to form a shroud segment. The plurality of shroud block assemblies includes a first shroud block assembly having a shroud block and a second shroud block assembly having a shroud block. The first shroud block assembly includes a seal interface member and a shroud seal. The seal interface member has a side portion that is adjacent to a radial side surface of the first shroud block. The second shroud block assembly includes a seal interface member and a shroud seal. The seal interface member has a side portion that is adjacent to a radial side surface of the second shroud block.