STM32F103 - Starting timers at the same time
The best way of doing this will be by using timer synchronization. See section 15.3.15 of the STM32F1 reference manual.

Set TIM3 to "Enable" master mode (MMS field of TIM3_CR2), and set TIM4 to "Trigger mode" slave mode with trigger ITR2 (TS and SMS fields of TIM4_SMCR). If you've configured everything properly, starting TIM3 will also simultaneously start TIM4.

(There is probably some way of doing this using the HAL, but I'm not familiar enough with it to say what that might be.)