Not really, they normally grow to 15 to 20m tall, and can reach a height of 30m. The corresponding spread could completely fill the 6m width of your garden.
My council put a silver birch in the grass verge outside of my house 30years ago and now is twice as tall as my house and the damn thing is a pain in the `fall ` when the leaves and twigs cover my lawn and gutters, so I would not advise anyone to grow one in such a small garden.