First off, be very wary of war baby stories, they are powerful bits of propaganda that are rarely true, in WW1 the British press was full of stories of German soldiers using Belgium and French babies as bayonet practice, and in the first gulf war who can forget about the babies being thrown out of incubators, it wasn’t true but it engenders good stories and further demonises the enemy.
Supporting the rights of Palestinians is not giving support to terrorists, their are huge civil rights abuses going on in the West Bank, just like you could sympathise with the plight of Catholics in Northern Ireland without supporting the IRA, and like it or not one man’s terrorist is another’s freedom fighter
This bloodshed is only a symptom of a festering wound that has not healed nor will it ever heal until the root causes are dealt with a fair and equitable solution until then and only then will peace prevail
As for the US position, it’s wholly for domestic consumption, but they have taken their eye off the ball, and allowed Israel to carry on with very little change