At the risk of straying into What If territory, I would say yes, the Allies (there were French troops present at Gallipoli as well) could have advanced further during the initial landing phase then they did. It is anybody's guess if the campaign could have actually achieved its objective of capturing Constantinople, many many things would have had to have been working in the Allied favour for this to happen.

While the landings were not made unopposed, many of the Turkish defenses were undermanned and poorly supplied. In the Helles sector where the majority of British forces landed, decent headway was made with some British forces coming within striking distance of the village of Krithia, one of the first day objectives. The village was undefended at the time and would have made a decent defensible position. This was reported the X beach but no response was forthcoming leading British commanders to order that their men consolidate the ground they had captured. Making matters worse, British troops, seeing boats evacuating the wounded, mistook this for a general re-embarkation and boarded the boats themselves thus withholding desperately needed reinforcements for those units who had repulsed Turkish counterattacks. These counterattacks eventually broke through British lines but the beachhead was saved by rallying British forces who forced the Turkish forces back. After this, a general re-embarkation was ordered during which no fire came from Turkish defenders and British Marines landed after the re-embarkation was completed faced no opposition, indicating the Turkish opposition had been largely broken by the British counterattack. Had the British commanders been better informed of the state of Turkish manpower, such an opportunity could have been exploited.

The situation at Anzac Cove was far worse. It is well known that the Australian and New Zealand forces were landed in the wrong place, instead of a gently sloping hill, the Australians and New Zealanders were faced with steep cliffs and ridges. While this point was relatively undefended, the terrain made any cohesive advance impossible and the battalions were mixed up amongst one another. Turkish resistance here was initially fairly light but again, the terrain made navigation difficult and what few defenders there were could use the terrain to their advantage. Slowed by the terrain and Turkish resistance, ANZAC forces were unable to make any significant headway before Turkish reinforcements arrived. By then, without room to manoeuvre, outflank or outgun the enemy, the Allied landing at ANZAC were doomed to failure. Since 1915 there has been significant criticism of the British for their incompetence during the Gallipoli campaign and while I don't think this is all unfounded, what the popular history neglects to mention is that the Australian and New Zealand forces were completely inexperienced, being all volunteer, and their Australian and New Zealand officers were in many cases just as incompetent through inexperience. More experienced forces and officers could have perhaps exploited the light Turkish defences despite the terrain but again, that is straying into What If territory.

Sources: Charles Bean: Official History of Australia in the War of 1914–1918 Les Carlyon: Gallipoli Peter Rees: Desert Boys