At a global level, yes, Islam is growing faster than Christianity: the 2016 Status of Global Mission reports Islam is growing globally at a rate of 1.87%, vs. Christianity's global growth rate of 1.3% and the population's global AGR of 1.1%. Bottom line: both Christianity and Islam are growing faster than the world population (so the world is becoming more religious); and Islam is growing faster than Christianity, so the % Muslim in the world is increasing.
BUT: total Christian adherents are significantly larger than total Muslims - 2.4 billion vs. 1.7 billion - and even by 2050, there will still be over a billion more Christians (3.4 billion) than Muslims (2.6 billion).
Islam's growth rate is higher in 93 countries. Christianity's growth rate is higher in 137 countries. (I am here comparing the AGR as given in the last edition of Operation World). Most of Islam's increase comes from demographic growth (births) in high-birth-rate countries where Islam is the majority. In countries where Christianity is the majority, the birth rate is lower (note: correlation, not causation). In most places, the twin benefit of demographics and conversion gives Christianity the edge. (Very little of Islam's growth rate comes from any kind of 'evangelical' conversion.)
A religion's growth rate tends to 'outstrip' the competing religion when the 'winner' is small. Smaller groups grow faster. For Islam, in the 10 countries where it has the highest % Muslim - Sahara, yemen, Morocco, Afghanistan, Muritania, Somalia, Tunisia, Maldives, Comoros, Iran - only in two (Somalia and Iran) is the Islamic AGR greater than the Christian AGR. BUT: the Christian population is a very tiny minority, and even with existing rapid rowth rates it will not overtake any time soon.
Islam's advantage comes in places where the birth rate is high, and some level of conflict or pressure leads to emigration of other religions. Examples include Niger, Djibouti, Turkey, Jordan, Turkmenistan, Iraq, Pakistan, TAjikistan, Kyrgyzstan, Qatar, Azerbaijan, Mali, Egypt, Uzbekistan.
In the largest populations of the world, it's a mixed bag:
So when someone says Islam is the fastest religion growing in the world, the answer is "Yes, but..." Let's not leap to the conclusion that Islam is the winner in terms of adherents. Christianity is larger in numbers globally, growing fastest where Islam is strongest, and certainly "in competition" in the largest populations.