Definitely impossible for most tracks. Possible for some, but at high costs. Germans have been upgrading some lines for decades now... The best (worst) example being the north-south towards Basel. For high speed the paths need to be further apart from eachother, there need to be better barriers between tracks and what's around them, the curves indeed need to be wider, the 'tilt' in the track in curves might need to be adjusted too... All of which leads to necessity of many new bridges and tunnels where this upgrading is impossible due to surroundings. It costs many millions of € per km and many decades to accomplish. The French on the other hand mainly went for "build new lines", it was clearly the better approach to get shit done fast (tho skipping many possible stops altogether on the new lines).

It's possible to upgrade a line like that, but it will involve a lot of cutting through hills or building bridges over valleys