Bleeding gums is a sign of gum disease and definitely not normal. One thing I would like to stress from the start is to NOT brush gentler, actually brush more! This is counterintuitive but brushing it more/better is what you should do.
Why does it bleed?
Bleeding occurs when you brush your teeth because your gums are inflamed. The inflammation is caused by the long term presence of bacteria that your gums react to. At spots that bleed, you will see that the gums are a darker red and may be slightly swollen. This is because the microscopic blood vessels in the gums are engorged. To even the gentlest touch, the unhealthy gums will bleed.
During pregnancy, it is common for gums to bleed. This is a special situation and is not necessarily a sign of unhealthy gums. It is the pregnancy hormones that increase the vascularity of the gums. Still brush as normal, as brushing gentler may actually leave plaque behind and worsen the bleeding.
How do I stop it from bleeding?
As we have established above, the gums bleed because of the chronic presence of bacteria. The most obvious thing to do is to remove the bacteria regularly, and that is the foundation to getting your gums healthy again. Unfortunately, there is no easy treatment that requires little effort from you.
There is no medication or mouthrinse to effectively solve the problem. It is all about physically removing the plaque everyday. This involves brushing really well and cleaning in between your teeth with floss, interdental brushes or waterpicks.
The bleeding is even worse when I brush harder!
Don't be afraid of the blood. With every other part of the body, if there is bleeding, we know to not disturb it and let it heal itself. However, with bleeding gums, it is the other way. It is very difficult to damage your gums with a toothbrush unless you are really trying to scrub your teeth. Plaque is soft and not stuck to teeth very strongly.
Imagine you are cleaning fresh (not dried) bolognese sauce off a plate. You don't need to scrub it really hard but still need a firm pressure.
Following that analogy, dental calculus is like baked on bolognese sauce. The kitchen sponge won't be able to get it off easily and something like steel wool might only work. This is similar to the scaling done by the dentist.
The worsened bleeding for the first couple of weeks of brushing better is perfectly fine. Keep it up and the gums will get healthier and bleed less.