Is toilet paper enough for istinja?

Yes, in most cases. Cleaning with toilet paper (called istijmar) is valid as long as it fully removes the impurity. The Prophet ﷺ allowed cleaning with stones, and paper does the same job. Water is better and gives a more thorough clean, and you need it if the mess spreads beyond the usual area.

What the Prophet ﷺ taught

Aisha reported that the Prophet ﷺ said: "When any of you goes to relieve himself, he should take with him three stones to clean himself, for they will be enough for him" (Sunan Abi Dawud 40). So dry cleaning was clearly allowed, and scholars of all four schools say that tissue or paper takes the place of stones.

He also used water. Anas ibn Malik said that he and another boy used to carry a tumbler of water for the Prophet ﷺ when he went to relieve himself (Sahih al-Bukhari 150). So both ways are part of the sunnah. The full overview is in a related article. See Do Muslims have to use water instead of toilet paper?

When paper alone is enough

The mess stays around the usual spot and has not spread to other areas.
You wipe at least three times, or more until the paper comes away clean. The Shafi'i and Hanbali schools require at least three wipes, while the Hanafi school requires only that you get clean. An odd number is recommended.
You use clean, pure paper or tissue that can absorb properly.
You use your left hand.

When water is the better choice

If it has spread beyond the usual area. The Shafi'i and Hanbali schools say that water is then required.
If paper can't get you clean, for example with loose stool.
If you want the cleanest result. The Quran praises people who love to purify themselves (Quran 9:108), and the verse is linked to the people of Quba, who used water after the toilet (Sunan Abi Dawud 44).

The best method is paper first, then water. At work or while travelling, a small bottle or a travel bidet makes this easy. Wet wipes work much like paper. See Can I use wet wipes for istinja?

What not to use

Salman reported that the Prophet ﷺ forbade cleaning with the right hand, with fewer than three stones, or with dung or bone (Sahih Muslim 262). Today, avoid using anything that is food or that may be harmful. Out of respect, avoid paper that carries Quran verses or the name of Allah.

Remember wudu

Cleaning yourself is not the same as wudu. Using the toilet breaks your wudu, so you must make wudu again before praying. The basics of istinja are explained separately. See What is istinja?

Key points

Toilet paper is valid for istinja if it removes the impurity completely (Sunan Abi Dawud 40).
Wipe at least three times, or until clean, with the left hand (Sahih Muslim 262).
Use water when the mess has spread or paper cannot clean you.
Paper followed by water is best.

Tip: Carry a small bottle of water in your bag if you are out and about. A quick rinse after paper means you never have to wonder if you are clean enough to pray.