We report a Yb-doped all-fiber laser system generating burst-mode pulses with high energy and high peak power at a GHz intra-burst repetition rate. To acquire the uniform burst envelope, a double-pre-compensation structure with an arbitrary waveform laser diode driver and an acoustic optical modulator is utilized for the first time. The synchronous pumping is utilized for the system to reduce the burst repetition rate to 100 Hz and suppress the amplified spontaneous emission effect. By adjusting the gain of every stage, uniform envelopes with different output energies can be easily obtained. The intra-burst repetition rate can be tuned from 0.5 to 10 GHz actively modulated by an electro-optic modulator. Optimized by timing control of eight channels of analog signal and amplified by seven stages of Yb-doped fiber amplifier, the pulse energy achieves 13.3 mJ at 0.5 ns intra-burst pulse duration, and the maximum peak power reaches approximately 3.6 MW at 48 ps intra-burst pulse duration. To the best of our knowledge, for reported burst-mode all-fiber lasers, this is a record for output energy and peak power with nanosecond-level burst duration, and the widest tuning range of the intra-burst repetition rate. In particular, this flexibly tunable burst-mode laser system can be directly applied to generate high-power frequency-tunable microwaves.