Is the word normalmente the same in Brazilian and European Portuguese?

Yes

The word "normalmente" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the final unstressed "e" is typically pronounced as a soft "i" (like the "ee" in "see"), and the rhythm is more syllable-timed. In Continental Portuguese, the unstressed vowels are often more closed or even elided (nearly silent) in natural speech, and the rhythm is more stress-timed.

Brazilian Portuguese Examples

1. Normalmente eu acordo cedo. (I usually wake up early.)
2. Ele normalmente janta tarde. (He normally eats dinner late.)
3. Normalmente não chove aqui nesta época. (It doesn't normally rain here at this time.)
4. A gente normalmente vai à praia no fim de semana. (We usually go to the beach on the weekend.)
5. Normalmente eu estudo à noite. (I normally study at night.)

Continental Portuguese Examples

1. Normalmente eu acordo cedo. (I usually wake up early.)
2. Ele costuma jantar tarde. (He usually eats dinner late.)
3. Normalmente não chove por aqui nesta época. (It doesn't normally rain around here at this time.)
4. Geralmente vamos à praia ao fim de semana. (Generally, we go to the beach at the weekend.)
5. Normalmente estudo à noite. (I normally study at night.)