Look at this chart. It is households consumption plus investment (there is no distinction between public and private investment). In real terms its level is still 6,6% below it peak  seen in the first quarter of 2008. It’s better than at the trough(first quarter of 2013) but it is still very weak.
If there is no recovery in the private internal demand, a strong and sustainable growth path is not attainable. (see here a development at the end of the post for the EA)

Two points to mention:
Its momentum is still too low to restore the GDP growth trajectory. As private debt is still high in many countries,it’s a drag to a strong recovery. (That’s a real difference with the US where households debt is now lower now than at the eve of the crisis) – For greater details on this point and on the role of private debt, the book from Atif Mian and Amir Sufi (House of Debt) is enlightening (see here).
The role of government expenditures has been important to support growth in recent years. Cutting them would reduce dramatically the demand addressed to companies leading then to a new recession.
As far as the private internal demand is not strong enough, cutting government expenditures could be dangerous, leading to weaker growth and higher unemployment. That’s something we have to keep in mind specifically in France.