No, not at all normal. If the bunny isn't producing any more normal round fecal balls and it's all pretty much that mushy mess, this is considered true diarrhea and is an immediate emergency. The risk is that it's being caused by pathogenic bacteria, which can prove fatal very quickly if the correct treatment isn't started immediately.
If the bunny is still producing plenty of normal round fecal balls, along with some mushy poop, the mushy part is likely mushy cecotropes. This isn't usually an emergency if corrected before it worsens to more serious digestive illness. The most common cause is sudden diet changes, introducing new foods too quickly, and/or introducing sugars and veggies into the diet before the bunny is old enough with a more developed digestive process. The stress of being in a new home can also be a contributing factor.
Is the bunny being fed the pellets and hay it had in it's previous home (if a new bunny to you), including being the exact same type and brand it's used to being fed? Any veggies, sugary foods, or high carb treats given? Any sudden changes to a baby rabbits diet or too many carbs, puts it at risk of developing enteritis.