Can someone really put a curse on me in Islam?

What Islam says is real

Islam does not dismiss magic or envy as fantasy. The Quran speaks of magic, and also puts its power in its place:

The Prophet ﷺ himself was affected by magic. He began to imagine that he had done something that he had not, until Allah told him the cause and cured him (Sahih al-Bukhari 5766). He also said, "The influence of an evil eye is a fact" (Sahih Muslim 2188). So harm from envy or sorcery is real, yet nothing happens outside Allah's will and decree.

Daily protection in Islam

The Prophet ﷺ taught simple habits that are far stronger than any fear:

• Morning and evening: say three times, "Bismillahilladhi la yadurru ma'asmihi shay'un fil-ardi wa la fis-sama'i wa huwas-Sami'ul-'Alim" (In the name of Allah, with whose name nothing on earth or in heaven can cause harm, and He is the Hearing, the Knowing). The Prophet ﷺ said nothing would harm the one who says it (Jami at-Tirmidhi 3388).
• After each prayer and before sleep: recite Ayat al-Kursi and the last three surahs: al-Ikhlas, al-Falaq and an-Nas. Surah al-Falaq asks Allah for protection from the evil of an envier.
• When you admire something: say "Masha'Allah" or "Barakallahu feek." See What is the evil eye in Islam?
• Keep your prayers, dua and charity. A heart close to Allah is a protected heart.

The full routine is covered here. See What are the morning and evening adhkar?

What not to do

• Do not go to a sorcerer or "healer" to remove a curse, and do not pay for cleansing rituals.
• Do not wear charms or knots for protection.
• Do not blame every problem on magic. Most illness, bad luck and relationship trouble has ordinary causes.

If you feel unwell, anxious or low, see a doctor first and use the Quran and dua alongside treatment, not instead of it. See Can jinn or black magic cause mental illness? For the wider topic of sorcery, see What is black magic in Islam?