I've noticed that your chances of taking a bridge without it being blown in your face increase by an enormous amount if you bombard/fire on the positions where the defenders are (or where you guess they will be, if you can't actually see them) well before you approach the bridge. In the HTTR scenarios, for example, you are almost guaranteed to take Grave bridge if you get Easy Company of 504 to fire on the bridge defenders, instead of trying to rush them, whilst at the same time bringing the 504 mortar platoon quickly into range and bombarding the defenders at the same time as easy company is firing - all this BEFORE your main body of 504 begins an attack. If you just give a battallion order to take the bridge then the bombard doesn't happen and 4 out of 5 times the bridge will blow in your face. Similarly, if you give the battallion that order but also give a separate, urgent take crossing order to Easy Co then 4 out of 5 times it goes bang.
So. In order to be successful, it seems, I really have to micro-manage this battallion attack a little. I have to give 3 separate orders. By itself the battallion will not plan an attack on the bridge which involves a prior bombard etc. Is this how the designers of the game envisaged play should be, I ask? It's not quite the complete hands-off orders-at-battallion-level play that the tutorial suggests, for example.