ATtiny84 resets on high load
Repost from the comments section:

Place a decoupling capacitor across the voltage input pins of your ATTiny. 100nF normally suffice to smoothen over short dips in the voltage that result from switching loads, starting motors etc. Try to place the capacitor as (physically) close to the pins as possible. You should have decoupling capacitors for all devices that keep some kind of state, e.g. memory banks, uCs and whatnot.