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

No.

While the meaning, spelling, and core grammar of the word evitar are identical in both dialects, there are differences in pronunciation and typical usage/syntax.

In Brazilian Portuguese, when expressing an ongoing action, the gerund form is used (evitando). In Continental Portuguese, the construction a + infinitive (a evitar) is the standard. Furthermore, pronunciation differs significantly: Brazilians pronounce the initial "e" clearly and openly, whereas Portuguese speakers often reduce the vowel, making it sound much more closed or almost swallowed.

Brazilian Portuguese Examples

1. Eu estou evitando comer carne ultimamente. (I am avoiding eating meat lately.) *2. Você precisa evitar esse caminho à noite. (You need to avoid this path at night.)
2. A gente tem que evitar confusão. (We have to avoid confusion.)
3. Ele estava evitando falar comigo. (He was avoiding speaking to me.)
4. Como você pode evitar o desperdício? (How can you avoid waste?)

Continental Portuguese Examples

1. Eu estou a evitar comer carne ultimamente. (I am avoiding eating meat lately.)
2. Tu precisas de evitar este caminho à noite. (You need to avoid this path at night.)
3. Temos de evitar a confusão. (We must avoid the confusion.)
4. Ele estava a evitar falar comigo. (He was avoiding speaking to me.)
5. Como podes evitar o desperdício? (How can you avoid waste?)