All agreed to his suggestion 【that】 a bridge 【be built】 across the river.是正确的.
知识点延伸：（1）agree to suggetion / plan / proposal同意建议/计划/意见；
（2）suggestion that 后面是一个同位语从句,补充说明suggestion的具体内容.suggestion做“建议”一义时,后面的同位语从句要用虚拟语气,谓语动词用should+动词原形,should可以省略.这里正是省略了should.
谨祝---------
好好学习,天天进步