I believe that it is impossible for freedom and equality not to conflict. The reason I say this is because there are so many different cultures in the world. Because of this there are multiple opinions, viewpoints, mores, customs, norms, beliefs, etc. Freedom gives an individual the right to form opinions about whomever they wish. Unfortunately, these opinions may be rooted in hatred or misunderstanding. They will naturally not think that everyone is equal to themselves for the simple fact that they believe in different things.
It would be nice if freedom and equality could exist in harmony but considering the diverse world we live in I think it is unlikely.
In a word, yes. There is no way that freedom and equality can keep from conflicting. (I guess they could if we were all perfect beings, but we're not.)
For example, let's say I want to use my freedom to abuse people from other races. Let's say I want to use my freedom to say that I will not give anyone who is not the same race as me a good grade in my class. There, freedom and equality conflict with each other.
On an economic level, they conflict as well. Let's say I want to use my money to pay for tutors and computer programs for my kid that will help her get smarter. A poor person can't do that. So my use of my freedom has made equality less likely.
They often tend to, but they can also coexist, or even have a cause-effect relationship.
Freedoms where humans are left to their own devices returns us to human nature, and the sometimes divisive, discriminating tendencies we have. We divide ourselves into rich and poor, dark and light, etc.
But freedom and equality coexist nicely at times as well. Take for example our freedom to worship, and inside most churches there is an equality in the eyes of God (or gods).
The freedoms we are given in the Bill of Rights, such as free speech or even the right to bear arms are freedoms given to everyone in the society, and they can cause equality between citizens too.
Freedom means not being in a position where you are forced by others to act in ways that you believe to be against your own interest.
Thus when a person takes up an employment and works hard for his employee to improve his future prospects, he still retains his freedom. But a slave who leads a comfortable life under a kind master is not free. So equality, or even inequality which is self imposed never causes loss of freedom. A mother may undergo untold amount of hardships and miseries to bring up a child under difficult economic situation. She may slog for the child much more than what even a slave or a paid employee is expected to do. This service to of mother for the child is not bondage. It is freedom.
Similarly when people accept equality that constraints their options, so that they can enjoy the benefits of cooperation and social law and orders, they are acting voluntarily in their own interest. Therefor this equality does not conflict with freedom.
However, if a thief insists on having the freedom to steal from others, and is put in jail, he or she has definitely lost his freedom because of the insistence of society on equality before law, and the equal application of social law prohibiting stealing.