Throw/put your weight behind something
Definition 1 of 1
Pro Tip 1/3Figurative Use
It's a metaphor that means lending support, not physically using your body weight. 
She threw her weight behind the new school policy.
Pro Tip 2/3Emphasizes Effort
The phrase suggests a strong and deliberate effort to support something. 
The community put their weight behind the fundraising event.
Pro Tip 3/3Shows Influence
Often implies that the person has the power or influence to make a difference. 
The celebrity threw his weight behind the charity, which helped raise awareness.