No it isn't normal. It also isn't common. Gun homicides don't even crack the top 10 causes of death in America, yet the BBC chooses to devote a disproportionate amount of their US front page to it. They certainly front-page shootings more than heart disease, that's the definition of bias.
Right or wrong, noble intentions or not, it's not an accurate portrayal of America as a whole on the part of the BBC. If CNN ran a front page column every time someone in the UK died of alcohol poisoning, would you consider that unbiased coverage of the UK?
There's no "normalization" - nobody treats murders as "normal". However, there's a difference between treating it as normal and treating is as bad, but rare abnormal, and putting it in the context of overall big picture. You can say "66 people are attacked by sharks recently" and it would look like a bloody carnage which warrants very grave concern. Or you can put it in the context that it's over all wide world, and there was one single shark attack fatality in the US over whole past year, and about 10x more people are killed each year by vending machines.
Right or wrong, noble intentions or not, it's not an accurate portrayal of America as a whole on the part of the BBC. If CNN ran a front page column every time someone in the UK died of alcohol poisoning, would you consider that unbiased coverage of the UK?