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Bridging the "terahertz (THz) gap" relies upon synthesizing arbitrary waveforms in the THz domain enabling applications that require both narrow band sources for sensing and few-cycle drives for classical and quantum objects. However, realization of custom-tailored waveforms needed for these applications is currently hindered due to limited flexibility for optical rectification of femtosecond pulses in bulk crystals. Here, we experimentally demonstrate that thin-film lithium niobate (TFLN) circuits provide a versatile solution for such waveform synthesis through combining the merits of complex integrated architectures, low-loss distribution of pump pulses on-chip, and an efficient optical rectification. Our distributed pulse phase-matching scheme grants shaping the temporal, spectral, phase, amplitude, and farfield characteristics of the emitted THz field through designer on-chip components. This strictly circumvents prior limitations caused by the phase-delay mismatch in conventional systems and relaxes the requirement for cumbersome spectral pre-engineering of the pumping light. We provide a toolbox of basic blocks that produce broadband emission up to 680 GHz with adaptable phase and coherence properties by using near-infrared pump pulse energies below 100 pJ.